m81304 Simplify and Use Square Roots By the end of this section, you will be able to: Simplify expressions with square roots Estimate square roots Approximate square roots Simplify variable expressions with square roots Use square roots in applications 36b90a09-411d-42af-8c04-3eaed7cb80ad Before you get started, take this readiness quiz. Simplify: (−9)2. If you missed this problem, review . 81 Round 3.846 to the nearest hundredth. If you missed this problem, review . 3.85 Evaluate 12d for d=80. If you missed this problem, review . 960

To start this section, we need to review some important vocabulary and notation. Remember that when a number n is multiplied by itself, we can write this as n2, which we read aloud as “nsquared.” For example, 82 is read as “8squared.” We call 64 the square of 8 because 82=64. Similarly, 121 is the square of 11, because 112=121. If n2=m, then m is the square of n.

Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine. This is why we say that the square of three is nine. 32=9 The number 9 is called a perfect square because it is the square of a whole number. Doing the Manipulative Mathematics activity Square Numbers will help you develop a better understanding of perfect square numbersThe chart shows the squares of the counting numbers 1 through 15. You can refer to it to help you identify the perfect squares. A perfect square is the square of a whole number. What happens when you square a negative number? (−8)2=(−8)(−8)=64 When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive. The chart shows the squares of the negative integers from −1 to −15. Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 102=100, we say 100 is the square of 10. We can also say that 10 is a square root of 100. A number whose square is m is called a square root of m. If n2=m, then n is a square root of m. Notice (−10)2=100 also, so −10 is also a square root of 100. Therefore, both 10 and −10 are square roots of 100. So, every positive number has two square roots: one positive and one negative.What if we only want the positive square root of a positive number? The radical sign, 0, stands for the positive square root. The positive square root is also called the principal square root. m is read as “the square root of m.” Ifm=n2,thenm=nforn≥0. We can also use the radical sign for the square root of zero. Because 02=0,0=0. Notice that zero has only one square root. The chart shows the square roots of the first 15 perfect square numbers. Simplify: ⓐ 25ⓑ 121. ⓐ 25 Since 52=25 5 ⓑ 121 Since 112=121 11 Simplify: ⓐ 36ⓑ 169. ⓐ 6 ⓑ 13 Simplify: ⓐ 16ⓑ 196. ⓐ 4 ⓑ 14 Every positive number has two square roots and the radical sign indicates the positive one. We write 100=10. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, −100=−10. Simplify. ⓐ −9ⓑ −144. ⓐ −9 The negative is in front of the radical sign. −3 ⓑ −144 The negative is in front of the radical sign. −12 Simplify: ⓐ −4ⓑ −225. ⓐ −2 ⓑ −15 Simplify: ⓐ −81ⓑ −64. ⓐ −9 ⓑ −8

Can we simplify −25? Is there a number whose square is −25? ()2=−25? None of the numbers that we have dealt with so far have a square that is −25. Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to −25. If we are asked to find the square root of any negative number, we say that the solution is not a real number. Simplify: ⓐ −169ⓑ −121. ⓐ There is no real number whose square is −169. Therefore, −169 is not a real number.ⓑ The negative is in front of the radical sign, so we find the opposite of the square root of 121. −121 The negative is in front of the radical. −11 Simplify: ⓐ −196ⓑ −81. ⓐ not a real number ⓑ −9 Simplify: ⓐ −49ⓑ −121. ⓐ not a real number ⓑ −11

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations. Simplify: ⓐ 25+144ⓑ 25+144. ⓐ Use the order of operations. 25+144 Simplify each radical. 5+12 Add. 17 ⓑ Use the order of operations. 25+144 Add under the radical sign. 169 Simplify. 13 Simplify: ⓐ 9+16ⓑ 9+16. ⓐ 7 ⓑ 5 Simplify: ⓐ 64+225ⓑ 64+225. ⓐ 17 ⓑ 23 Notice the different answers in parts ⓐ and ⓑ of . It is important to follow the order of operations correctly. In ⓐ , we took each square root first and then added them. In ⓑ , we added under the radical sign first and then found the square root.

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers. We might conclude that the square roots of numbers between 4 and 9 will be between 2 and 3, and they will not be whole numbers. Based on the pattern in the table above, we could say that 5 is between 2 and 3. Using inequality symbols, we write 2<5<3 Estimate 60 between two consecutive whole numbers. Think of the perfect squares closest to 60. Make a small table of these perfect squares and their squares roots. Locate 60 between two consecutive perfect squares. 49<60<64 60is between their square roots. 7<60<8 Estimate 38 between two consecutive whole numbers. 6<38<7 Estimate 84 between two consecutive whole numbers. 9<84<10

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the 0 or x key on your calculator. You will need to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is ≈ and it is read approximately. Suppose your calculator has a 10-digit display. Using it to find the square root of 5 will give 2.236067977. This is the approximate square root of 5. When we report the answer, we should use the “approximately equal to” sign instead of an equal sign. 5≈2.236067978 You will seldom use this many digits for applications in algebra. So, if you wanted to round 5 to two decimal places, you would write 5≈2.24 How do we know these values are approximations and not the exact values? Look at what happens when we square them. 2.2360679782=5.0000000022.242=5.0176The squares are close, but not exactly equal, to 5. Round 17 to two decimal places using a calculator. 17 Use the calculator square root key. 4.123105626 Round to two decimal places. 4.12 17≈4.12 Round 11 to two decimal places. ≈ 3.32 Round 13 to two decimal places. ≈ 3.61

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression? Consider 9x2, where x≥0. Can you think of an expression whose square is 9x2? (?)2=9x2(3x)2=9x2so9x2=3xWhen we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero. Simplify: x2. Think about what we would have to square to get x2. Algebraically, (?)2=x2 x2 Since (x)2=x2 x Simplify: y2. y Simplify: m2. m Simplify: 16x2. 16x2 Since(4x)2=16x2 4x Simplify: 64x2. 8x Simplify: 169y2. 13y Simplify: −81y2. −81y2 Since(9y)2=81y2 −9y Simplify: −121y2. −11y Simplify: −100p2. −10p Simplify: 36x2y2. 36x2y2 Since(6xy)2=36x2y2 6xy Simplify: 100a2b2. 10ab Simplify: 225m2n2. 15mn

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer! Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Write a complete sentence that answers the question.

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area. If the area of the square is A square units, the length of a side is A units. See . Area (square units) Length of side (units) 9 9=3 144 144=12 A A Mike and Lychelle want to make a square patio. They have enough concrete for an area of 200 square feet. To the nearest tenth of a foot, how long can a side of their square patio be? We know the area of the square is 200 square feet and want to find the length of the side. If the area of the square is A square units, the length of a side is A units. What are you asked to find? The length of each side of a square patio Write a phrase. The length of a side Translate to an expression. A Evaluate A when A=200. 200 Use your calculator. 14.142135... Round to one decimal place. 14.1 feet Write a sentence. Each side of the patio should be 14.1 feet. Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. To the nearest tenth of a foot, how long can a side of her square lawn be? 19.2 feet Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. How long can a side of his mosaic be? 52 centimeters

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by evaluating the expression h4. For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by evaluating 644. 644 Take the square root of 64. 84 Simplify the fraction. 2 It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground. Christy dropped her sunglasses from a bridge 400 feet above a river. How many seconds does it take for the sunglasses to reach the river? What are you asked to find? The number of seconds it takes for the sunglasses to reach the river Write a phrase. The time it will take to reach the river Translate to an expression. h4 Evaluate h4 when h=400. 4004 Find the square root of 400. 204 Simplify. 5 Write a sentence. It will take 5 seconds for the sunglasses to reach the river. A helicopter drops a rescue package from a height of 1296 feet. How many seconds does it take for the package to reach the ground? 9 seconds A window washer drops a squeegee from a platform 196 feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk? 3.5 seconds

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is d feet, then the speed of the car can be found by evaluating 24d. After a car accident, the skid marks for one car measured 190 feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied? What are you asked to find? The speed of the car before the brakes were applied Write a phrase. The speed of the car Translate to an expression. 24d Evaluate24dwhend=190. 24·190 Multiply. 4,560 Use your calculator. 67.527772... Round to tenths. 67.5 Write a sentence. The speed of the car was approximately 67.5 miles per hour. An accident investigator measured the skid marks of a car and found their length was 76 feet. To the nearest tenth, what was the speed of the car before the brakes were applied? 42.7 mph The skid marks of a vehicle involved in an accident were 122 feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied? 54.1 mph The Links to Literacy activity "Sea Squares" will provide you with another view of the topics covered in this section. Introduction to Square Roots Estimating Square Roots with a Calculator

Square Root Notation m is read ‘the square root of m’ If m=n2, then m=n, for n≥0. Use a strategy for applications with square roots. Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Write a complete sentence that answers the question.

Simplify Expressions with Square Roots In the following exercises, simplify. 36 6 4 64 8 144 −4 −2 −100 −1 −1 −121 −121 not a real number −36 −9 not a real number −49 9+16 5 25+144 9+16 7 25+144 Estimate Square Roots In the following exercises, estimate each square root between two consecutive whole numbers. 70 8<70<9 55 200 14<200<15 172 Approximate Square Roots with a Calculator In the following exercises, use a calculator to approximate each square root and round to two decimal places. 19 4.36 21 53 7.28 47 Simplify Variable Expressions with Square Roots In the following exercises, simplify. (Assume all variables are greater than or equal to zero.) y2 y b2 49x2 7x 100y2 −64a2 −8a −25x2 144x2y2 12xy 196a2b2 Use Square Roots in Applications In the following exercises, solve. Round to one decimal place. Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. How long can a side of his garden be? 8.7 feet Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. How long can a side of his patio be? Gravity An airplane dropped a flare from a height of 1,024 feet above a lake. How many seconds did it take for the flare to reach the water? 8 seconds Gravity A hang glider dropped his cell phone from a height of 350 feet. How many seconds did it take for the cell phone to reach the ground? Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4,000 feet above the Colorado River. How many seconds did it take for the hammer to reach the river? 15.8 seconds Accident investigation The skid marks from a car involved in an accident measured 54 feet. What was the speed of the car before the brakes were applied? Accident investigation The skid marks from a car involved in an accident measured 216 feet. What was the speed of the car before the brakes were applied? 72 mph Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. What was the speed of the vehicle before the brakes were applied? Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. What was the speed of the vehicle before the brakes were applied? 53.0 mph

Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be? Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2,025 tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be? 45 inches

Why is there no real number equal to −64? What is the difference between 92 and 9? Answers will vary. 9² reads: “nine squared” and means nine times itself. The expression 9 reads: “the square root of nine” which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Name Decimals In the following exercises, name each decimal. 0.8 0.375 three hundred seventy-five thousandths 0.007 5.24 five and twenty-four hundredths −12.5632 −4.09 negative four and nine hundredths Write Decimals In the following exercises, write as a decimal. three tenths nine hundredths 0.09 twenty-seven hundredths ten and thirty-five thousandths 10.035 negative twenty and three tenths negative five hundredths −0.05 Convert Decimals to Fractions or Mixed Numbers In the following exercises, convert each decimal to a fraction. Simplify the answer if possible. 0.43 0.825 3340 9.7 3.64 31625 Locate Decimals on the Number Line ⓐ 0.6ⓑ −0.9ⓒ 2.2ⓓ −1.3 Order Decimals In the following exercises, order each of the following pairs of numbers, using < or >. 0.6___0.8 < 0.2___0.15 0.803____0.83 < −0.56____−0.562 Round Decimals In the following exercises, round each number to the nearest: ⓐ hundredth ⓑ tenth ⓒ whole number. 12.529 ⓐ 12.53 ⓑ 12.5 ⓒ 13 4.8447 5.897 ⓐ 5.90 ⓑ 5.9 ⓒ 6

Add and Subtract Decimals In the following exercises, add or subtract. 5.75+8.46 32.89−8.22 24.67 24−19.31 10.2+14.631 24.831 −6.4+(−2.9) 1.83−4.2 −2.37 Multiply Decimals In the following exercises, multiply. (0.3)(0.7) (−6.4)(0.25) −1.6 (−3.35)(−12.7) (15.4)(1000) 15,400 Divide Decimals In the following exercises, divide. 0.48÷6 4.32÷24 0.18 $6.29÷12 (−0.8)÷(−0.2) 4 1.65÷0.15 9÷0.045 200 Use Decimals in Money Applications In the following exercises, use the strategy for applications to solve. Miranda got $40 from her ATM. She spent $9.32 on lunch and $16.99 on a book. How much money did she have left? Round to the nearest cent if necessary. Jessie put 8 gallons of gas in her car. One gallon of gas costs $3.528. How much did Jessie owe for all the gas? $28.22 A pack of 16 water bottles cost $6.72. How much did each bottle cost? Alice bought a roll of paper towels that cost $2.49. She had a coupon for $0.35 off, and the store doubled the coupon. How much did Alice pay for the paper towels? $1.79

Convert Fractions to Decimals In the following exercises, convert each fraction to a decimal. 35 78 0.875 −1920 −214 −5.25 13 611 0.54— Order Decimals and Fractions In the following exercises, order each pair of numbers, using < or >. 12___0.2 35___0. > −78___−0.84 −512___−0.42 > 0.625___1320 0.33___516 > In the following exercises, write each set of numbers in order from least to greatest. 23,1720,0.65 79,0.75,1115 1115,0.75,79 Simplify Expressions Using the Order of Operations In the following exercises, simplify 4(10.3−5.8) 34(15.44−7.4) 6.03 30÷(0.45+0.15) 1.6+38 1.975 52(0.5)+(0.4)2 −25·910+0.14 −0.22 Find the Circumference and Area of Circles In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle. radius=6 in. radius=3.5 ft. ⓐ 21.98 ft. ⓑ 38.465 sq.ft. radius=733m diameter=11 cm ⓐ 34.54 cm ⓑ 94.985 sq.cm

Determine Whether a Decimal is a Solution of an Equation In the following exercises, determine whether the each number is a solution of the given equation. x−0.4=2.1 ⓐ x=1.7 ⓑ x=2.5 y+3.2=−1.5ⓐ y=1.7ⓑ y=−4.7 ⓐ no ⓑ yes u2.5=−12.5ⓐ u=−5ⓑ u=−31.25 0.45v=−40.5ⓐ v=−18.225ⓑ v=−90 ⓐ no ⓑ yes Solve Equations with Decimals In the following exercises, solve. m+3.8=7.5 h+5.91=2.4 h = −3.51 a+2.26=−1.1 p−4.3=−1.65 p = 2.65 x−0.24=−8.6 j−7.42=−3.7 j = 3.72 0.6p=13.2 −8.6x=34.4 x = −4 −22.32=−2.4z a0.3=−24 a = −7.2 p−7=−4.2 s−2.5=−10 s = 25 Translate to an Equation and Solve In the following exercises, translate and solve. The difference of n and 15.2 is 4.4. The product of −5.9 and x is −3.54. −5.9x = −3.54; x = 0.6 The quotient of y and −1.8 is −9. The sum of m and (−4.03) is 6.8. m + (−4.03) = 6.8; m = 10.83

Find the Mean of a Set of Numbers In the following exercises, find the mean of the numbers. 2,4,1,0,1,and1 $270, $310.50, $243.75, and $252.15 $269.10 Each workday last week, Yoshie kept track of the number of minutes she had to wait for the bus. She waited 3,0,8,1,and8 minutes. Find the mean. In the last three months, Raul’s water bills were $31.45,$48.76,and$42.60. Find the mean. $40.94 Find the Median of a Set of Numbers In the following exercises, find the median. 41, 45, 32, 60, 58 25, 23, 24, 26, 29, 19, 18, 32 24.5 The ages of the eight men in Jerry’s model train club are 52,63,45,51,55,75,60,and59. Find the median age. The number of clients at Miranda’s beauty salon each weekday last week were 18,7,12,16,and20. Find the median number of clients. 16 clients Find the Mode of a Set of Numbers In the following exercises, identify the mode of the numbers. 6, 4, 4,5, 6,6, 4, 4, 4, 3, 5 The number of siblings of a group of students: 2, 0, 3, 2, 4, 1, 6, 5, 4, 1, 2, 3 2 Use the Basic Definition of Probability In the following exercises, solve. (Round decimals to three places.) The Sustainability Club sells 200 tickets to a raffle, and Albert buys one ticket. One ticket will be selected at random to win the grand prize. Find the probability Albert will win the grand prize. Express your answer as a fraction and as a decimal. Luc has to read 3 novels and 12 short stories for his literature class. The professor will choose one reading at random for the final exam. Find the probability that the professor will choose a novel for the final exam. Express your answer as a fraction and as a decimal. 15;0.2

Write a Ratio as a Fraction In the following exercises, write each ratio as a fraction. Simplify the answer if possible. 28 to 40 56 to 32 74 3.5 to 0.5 1.2 to 1.8 23 134to158 213to514 49 64 ounces to 30 ounces 28 inches to 3 feet 79 Write a Rate as a Fraction In the following exercises, write each rate as a fraction. Simplify the answer if possible. 180 calories per 8 ounces 90 pounds per 7.5 square inches 12pounds1square inch 126 miles in 4 hours $612.50 for 35 hours $352hours Find Unit Rates In the following exercises, find the unit rate. 180 calories per 8 ounces 90 pounds per 7.5 square inches 12 pounds/sq.in. 126 miles in 4 hours $612.50 for 35 hours $17.50/hour Find Unit Price In the following exercises, find the unit price. t-shirts: 3 for $8.97 Highlighters: 6 for $2.52 $0.42 An office supply store sells a box of pens for $11. The box contains 12 pens. How much does each pen cost? Anna bought a pack of 8 kitchen towels for $13.20. How much did each towel cost? Round to the nearest cent if necessary. $1.65 In the following exercises, find each unit price and then determine the better buy. Shampoo: 12 ounces for $4.29 or 22 ounces for $7.29? Vitamins: 60 tablets for $6.49 or 100 for $11.99? $0.11, $0.12; 60 tablets for $6.49 Translate Phrases to Expressions with Fractions In the following exercises, translate the English phrase into an algebraic expression. 535 miles per hhours a adults to 45 children aadults45children the ratio of 4y and the difference of x and 10 the ratio of 19 and the sum of 3 and n 193+n

Simplify Expressions with Square Roots In the following exercises, simplify. 64 144 12 −25 −81 −9 −9 −36 not a real number 64+225 64+225 17 Estimate Square Roots In the following exercises, estimate each square root between two consecutive whole numbers. 28 155 12<155<13 Approximate Square Roots In the following exercises, approximate each square root and round to two decimal places. 15 57 7.55 Simplify Variable Expressions with Square Roots In the following exercises, simplify. (Assume all variables are greater than or equal to zero.) q2 64b2 8b −121a2 225m2n2 15mn −100q2 49y2 7y 4a2b2 121c2d2 11cd Use Square Roots in Applications In the following exercises, solve. Round to one decimal place. Art Diego has 225 square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be? Landscaping Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of 30 square feet. How long can a side of the flower garden be? 5.5 feet Gravity A hiker dropped a granola bar from a lookout spot 576 feet above a valley. How long did it take the granola bar to reach the valley floor? Accident investigation The skid marks of a car involved in an accident were 216 feet. How fast had the car been going before applying the brakes? 72 mph

Write six and thirty-four thousandths as a decimal. Write 1.73 as a fraction. 173100 Write 58 as a decimal. Round 16.749 to the nearest ⓐ tenth ⓑ hundredth ⓒ whole number ⓐ 16.7 ⓑ 16.75 ⓒ 17 Write the numbers 45,−0.1,0.804,29,−7.4,0.21 in order from smallest to largest. In the following exercises, simplify each expression. 15.4+3.02 18.42 20−5.71 (0.64)(0.3) 0.192 (−4.2)(100) 0.96÷(−12) −0.08 −5÷0.025 −0.6÷(−0.3) 2 (0.7)2 24÷(0.1+0.02) 200 4(10.3−5.8) 1.6+38 1.975 23(14.65−4.6) In the following exercises, solve. m+3.7=2.5 −1.2 h0.5=4.38 −6.5y=−57.2 8.8 1.94=a−2.6 Three friends went out to dinner and agreed to split the bill evenly. The bill was $79.35. How much should each person pay? $26.45 A circle has radius 12. Find the ⓐ circumference and ⓑ area. [Use3.14forπ.] The ages, in months, of 10 children in a preschool class are: 55, 55, 50, 51, 52, 50, 53, 51, 55, 49 Find the ⓐ mean ⓑ median ⓒ mode ⓐ 52.1 ⓑ 51.5 ⓒ 55 Of the 16 nurses in Doreen’s department, 12 are women and 4 are men. One of the nurses will be assigned at random to work an extra shift next week. ⓐ Find the probability a woman nurse will be assigned the extra shift. ⓑ Convert the fraction to a decimal. Find each unit price and then the better buy. Laundry detergent: 64 ounces for $10.99 or 48 ounces for $8.49 The unit prices are $0.172 per ounce for 64 ounces, and $0.177 per ounce for 48 ounces; 64 ounces is the better buy. In the following exercises, simplify. 36+64 144n2 12n Estimate 54 to between two whole numbers. Yanet wants a square patio in her backyard. She has 225 square feet of tile. How long can a side of the patio be? 15 feet