{
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  "name": "",
  "signature": "sha256:e0b439f2a97b0911a972c7ec62c1840ed5b08a289ea36787dc6abfb4315c683d"
 },
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Advanced Expression Manipulation Solutions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "x, y, z = symbols('x y z')\n",
      "init_printing()\n",
      "from IPython.display import display"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For each exercise, fill in the function according to its docstring. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Creating expressions from classes"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Create the following objects without using any mathematical operators like `+`, `-`, `*`, `/`, or `**` by explicitly using the classes `Add`, `Mul`, and `Pow`.  You may use `x` instead of `Symbol('x')` and `4` instead of `Integer(4)`.\n",
      "\n",
      "$$x^2 + 4xyz$$\n",
      "$$x^{(x^y)}$$\n",
      "$$x - \\frac{y}{z}$$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def explicit_classes1():\n",
      "    \"\"\"\n",
      "    Returns the expression x**2 + 4*x*y*z, built using SymPy classes explicitly.\n",
      "\n",
      "    >>> explicit_classes1()\n",
      "    x**2 + 4*x*y*z\n",
      "    \"\"\"\n",
      "    return Add(Pow(x, 2), Mul(4, x, y, z))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "explicit_classes1()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} + 4 x y z$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAYBAMAAABjIUxVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 36,
       "text": [
        " 2          \n",
        "x  + 4\u22c5x\u22c5y\u22c5z"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def explicit_classes2():\n",
      "    \"\"\"\n",
      "    Returns the expression x**(x**y), built using SymPy classes explicitly.\n",
      "\n",
      "    >>> explicit_classes2()\n",
      "    x**(x**y)\n",
      "    \"\"\"\n",
      "    return Pow(x, Pow(x, y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "explicit_classes2()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{x^{y}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAUBAMAAABscEDKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIol2q1SZEGbd7zK7\nRM2RoJmtAAAAlUlEQVQYGWNgQIBgBT4Eh4GhN2ELMpdlgwUyl5PhNIyr7Guygdl4AZTLVODDFSDB\nqgDl8jK8YGDY1gzlAamPCCaQxb2AG4m/n+kBExK3xjRZHcIVMhE1M4DLsCtwzOCaAOcyMTB/YN0A\n57IzsByAc0AMLoROEFd+A0MBiAYDDgF7BmYEN77hFIMwTI6BocaoSLkBzgUAQBUYbmElZeQAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 38,
       "text": [
        " \u239b y\u239e\n",
        " \u239dx \u23a0\n",
        "x    "
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def explicit_classes3():\n",
      "    \"\"\"\n",
      "    Returns the expression x - y/z, built using SymPy classes explicitly.\n",
      "\n",
      "    >>> explicit_classes3()\n",
      "    x - y/z\n",
      "    \"\"\"\n",
      "    return Add(x, Mul(-1, Mul(y, Pow(z, -1))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "explicit_classes3()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x - \\frac{y}{z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAAlBAMAAAAdAMyfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIol2q1SZEGbd7zK7\nzUTvhYErAAAA7UlEQVQoFWNgwALY1RQY5AuwSDAwMPF9YLDHKsOwlW0CwxnsUgVcBgyfsUsxxBdw\n/8YhlcvAvACH1DUGHgccUikM7xtwSNUY5wvgkGJg0MQhwzKBfSkOKU4FHgUcUqzGeThk4MJCJqJm\nBnAeMoNdgWMG1wRkETibiYH5A+sGOBeZwc7AcgDMl50JAvNA7P8g8AHIAMYALiC/gaEAqxyHgD0D\nM3ap+IZTDMJYNTHUGBUpN2CXGsyifKse4PAQg0xz2gEcTi9gKMEeQED1jA44NDEwPMMpw6qAUyqI\ngcEBuyT3jYzeB9il2Bj6gakXAFr6LczlCLDrAAAAAElFTkSuQmCC\n",
       "prompt_number": 40,
       "text": [
        "    y\n",
        "x - \u2500\n",
        "    z"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Nested args"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = x**2 - y*(2**(x + 3) + z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use nested `.args` calls to get the 3 in expr."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def nested_args():\n",
      "    \"\"\"\n",
      "    Get the 3 in the above expression.\n",
      "\n",
      "    >>> nested_args()\n",
      "    3\n",
      "    \"\"\"\n",
      "    expr = x**2 - y*(2**(x + 3) + z)\n",
      "    return expr.args[0].args[2].args[1].args[1].args[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "nested_args()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMom7VCKZRHbdzavv\nEGbh6M2uAAAATklEQVQIHWNgEFIyYWBgTWCob2Bg+sLAP4GBBUwyMIBEgGA1EDNqbQAxGXoKQCTP\nXwZWAQbmfwz3/4JI/gUMnF8ZOB4w1CswMLilLWEAAJ6qEfrOOFffAAAAAElFTkSuQmCC\n",
       "prompt_number": 43,
       "text": [
        "3"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Traversal "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Write a post-order traversal function that prints each node."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def post(expr):\n",
      "    \"\"\"\n",
      "    Post-order traversal\n",
      "\n",
      "    >>> expr = x**2 - y*(2**(x + 3) + z)\n",
      "    >>> post(expr)\n",
      "    -1\n",
      "    y\n",
      "    2\n",
      "    3\n",
      "    x\n",
      "    x + 3\n",
      "    2**(x + 3)\n",
      "    z\n",
      "    2**(x + 3) + z\n",
      "    -y*(2**(x + 3) + z)\n",
      "    x\n",
      "    2\n",
      "    x**2\n",
      "    x**2 - y*(2**(x + 3) + z)\n",
      "    \"\"\"\n",
      "    for arg in expr.args:\n",
      "        post(arg)\n",
      "    display(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = x**2 - y*(2**(x + 3) + z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "post(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-1$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMAIs3dRLsQq3aZ72YDTSHvAAAALklE\nQVQIHWNggIMwOIuBIfUYgpNejsRh4CCTI2QMAiYMZBsAcxDnrEOzYWwwDQBP9Q53CSxxJQAAAABJ\nRU5ErkJggg==\n",
       "text": [
        "-1"
       ]
      },
      {
       "latex": [
        "$$y$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAANBAMAAACN24kIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZIkR2MlTdu+/N\nq2Z6CBlbAAAAWElEQVQIHWNgVFZg8BdgYGL9wJDPwFDBPIFhFQODAHcCwzcGBoZ+AfbvQMqGgW0D\nkDrMwHUBSNkyxD8AUlJp9g5AioFBE4h5JzDuAFKcClwKQIojzRRIAgDYaQ5MI2ffIgAAAABJRU5E\nrkJggg==\n",
       "text": [
        "y"
       ]
      },
      {
       "latex": [
        "$$z$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAJBAMAAAD5iKAgAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMARO8idrtmmRCrVIkyzciQ\n8DMAAAA4SURBVAgdY2BgVBIwYWAwTSkPYGBoYOhsYGBgYJ4AJBjEQQTXBhC5moFhAgPPtbJcAQZ2\nhjwtBgDU0whkAL7DjgAAAABJRU5ErkJggg==\n",
       "text": [
        "z"
       ]
      },
      {
       "latex": [
        "$$2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAAAT0lEQVQIHWNgVDIJYGAQY2D/wcCQycCwnoFh9wSG/AAG+wkM+kAJoMgEIMHzF8Rk\ndgCRKiCCPQFENjEwCjDwKDDwCTBI3b2rBVT8//8vBgBR7Q/Fm5CwxAAAAABJRU5ErkJggg==\n",
       "text": [
        "2"
       ]
      },
      {
       "latex": [
        "$$3$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMom7VCKZRHbdzavv\nEGbh6M2uAAAATklEQVQIHWNgEFIyYWBgTWCob2Bg+sLAP4GBBUwyMIBEgGA1EDNqbQAxGXoKQCTP\nXwZWAQbmfwz3/4JI/gUMnF8ZOB4w1CswMLilLWEAAJ6qEfrOOFffAAAAAElFTkSuQmCC\n",
       "text": [
        "3"
       ]
      },
      {
       "latex": [
        "$$x$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAIol2q1SZEGbd7zK78ZNZ0gAA\nAEJJREFUCB1jYBAyETUzYGBX4JjBNYGBiYH5AusGBnYGlgMMIMBlAKZkNzAUMHAI2DIwFzDENpxi\nEGZgqDEqUm5gAAAhlwo0cLKzVgAAAABJRU5ErkJggg==\n",
       "text": [
        "x"
       ]
      },
      {
       "latex": [
        "$$x + 3$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAQBAMAAAB0JTvnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIol2q1SZEGbd7zK7\nzUTvhYErAAAAvUlEQVQYGWNgwATcSjoBmKJAETEGvs8oEuwCEK5+AMNPFAlWAwg3HpcEA8QoIRNR\nM4hSmA4GBl4HBgZ2BY4ZXBPAZsAlNl0B8pkYmD+wbkCVYGBeB9TBwHIALMzAANfBwLC+ACjGBbaB\n7e7dm2vv3nUACkgzMJxvANLyGxhA8nAd7P8LQBIcAvYMzCgSDH8ZGOYLMMQ3nGIQBmuA22HBwPeJ\ngaHGqEi5AVWCNS03ACICJZFchSLOwAd2HlQMAHhUI/VdUwmwAAAAAElFTkSuQmCC\n",
       "text": [
        "x + 3"
       ]
      },
      {
       "latex": [
        "$$2^{x + 3}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAUBAMAAAD8YzkFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAAA1UlEQVQYGWNgQANzazNRRRqAXD0GZVRBByA3ieE9piADw2WoIG9ZeA6Q6QDicm4F\nkUBwk81hB5ByAGLOkgsMDIxKJgEMAcwPgHyISgYVBgYxBvYfDAz8Agxsxsabjc2BUvINDEBXrWdg\nOM1wAKKS8SuD/AOG3RMY8mM+XGYAmgTSzrmQ4fUEBvsJDPqR7k21EyCCDMdrE4EsoHaQAAg4gEkw\nwfMXxk6AMRgYmB0QbDgL6C4MwJ6AIcTA0MTAKIAuzKPAwIchKHX3rha6Qob1////QhcEANeKK2sF\nlvrRAAAAAElFTkSuQmCC\n",
       "text": [
        " x + 3\n",
        "2     "
       ]
      },
      {
       "latex": [
        "$$2^{x + 3} + z$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAWBAMAAACYrGeJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAABRElEQVQoFWNgQANzazPRRDC4DUARPQZlDHE0AQcgP4nhPZooBhekjIHhMpjERvCW\nhecAxR1AcpxbQSRWcJPNYQdQwgGIOUsuYFPCqGQSwBDA/AAk5wAiGFTAJBohxsD+g4GBX4CBzdh4\ns7E5UFa+AU0JiAsMpfUMDKcZDgDZDgwMjF8Z5B8AmUAwEUJByN0TGPJjPlxmALnIAei0hQyvJ0Bk\n2iEUhLSfwKAf6d5UC5JzAOLjtYkQCQYUZUCx9VDtUC9AVcGVse4OEAeK8fyFSSTAGGAaZprIcyUH\noAAziMACYMomMEwDWYc1nIDiMGUMjAlAHjuIwADKxsarjY3NwOJBILKJgVEAQxVIAGYaxwUgh0eB\ngQ+/sl4GhgQGqbt3tbAaBjONx07vcQDD+v//f+FVxsTwHpR6cAKY23AqgEiEI8kDANtZPfbhKwet\nAAAAAElFTkSuQmCC\n",
       "text": [
        " x + 3    \n",
        "2      + z"
       ]
      },
      {
       "latex": [
        "$$- y \\left(2^{x + 3} + z\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAcBAMAAABRxMbXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIs3dRBCJmXYyVLvv\nq2aiBbi/AAACQ0lEQVQ4EZWTMWgUQRSG/93s3u7esZdDGyFKjhVBENSgCFbZRtTKrVIpWQKmPgx6\nhRCvuhQaCNhYKG5lLK+SNIIEC5EUJ6Ji5WElKOREI2hhfG9m5273dkjOV8y8/3vvn9mZuQPGiLn2\nuzG6tC0J0WXc0Na00AqzuEPiA7azSOWxSnLzo5xiO7CQY6nIN6bQPiaT0uqlj5QJu/0yLeYmt5WT\nUvg9OV/3O68pY7t9N5ZsZJwZ0SwPA0bzRIjQ7LGUH3+b00KsFgjQBI7A/QNM1uAHwWZwnJqmE00n\n5qMipWPSK88CD7FGVdrd+Inpnmy0a1nDREozzOsDWxE+Xe4vICZOdnsG3yLZUm7JWY4OFUfCfQGc\ninDnwsZ6mz3c8aD9Ju3K26v9FANzt2DskPLrAs2ylYPtw1B2ays8BFhskBFfgcuLVbqsvV+SAkIq\nAWWf+tKkdb3fquAlS6gQgdlgZHKqCWWPcC0Sd3rgDMdZG68w2SLDhLDr3xmD3WF0qdem502jvIPH\na5QLu8vFQjhBcHQzCETtIlfpRVU4ddzj3OTqOoyaKuRm9fHlmLExODuf+Tkjv0NX0kB1b/sTcafW\nd3aIcLrWX07c08DBxcUlSUfHdHfv5PLnEMi8u72xItay+vSL3d39MWqUOrU72OZ/Y6me6SrRvhTD\n82RqKlVnl9psKY63uJoI8X6ANInVysL54QWdwzNZuZ9t2Dv/Oiw/XQmlqCRDuE92XlM36hqoRdWG\nDtObjhdT2rZqosVFeLOI/pP8A3zcd135cM5RAAAAAElFTkSuQmCC\n",
       "text": [
        "   \u239b x + 3    \u239e\n",
        "-y\u22c5\u239d2      + z\u23a0"
       ]
      },
      {
       "latex": [
        "$$x$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAIol2q1SZEGbd7zK78ZNZ0gAA\nAEJJREFUCB1jYBAyETUzYGBX4JjBNYGBiYH5AusGBnYGlgMMIMBlAKZkNzAUMHAI2DIwFzDENpxi\nEGZgqDEqUm5gAAAhlwo0cLKzVgAAAABJRU5ErkJggg==\n",
       "text": [
        "x"
       ]
      },
      {
       "latex": [
        "$$2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAAAT0lEQVQIHWNgVDIJYGAQY2D/wcCQycCwnoFh9wSG/AAG+wkM+kAJoMgEIMHzF8Rk\ndgCRKiCCPQFENjEwCjDwKDDwCTBI3b2rBVT8//8vBgBR7Q/Fm5CwxAAAAABJRU5ErkJggg==\n",
       "text": [
        "2"
       ]
      },
      {
       "latex": [
        "$$x^{2}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAUBAMAAAB/pwA+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIol2q1SZEGbd7zK7\nzUTvhYErAAAAgUlEQVQIHWNggAC1IwVQFlsB2wUok1OA8TeUyRXA/g/KZGBg/Q5n8irAmSpwFgdC\n8BHDbqgw99FQdyiT6///DwwMQiaiZgZgEXYFjhlcE8BMJgbmD6wbIKIMLAfADBDBBVEJYspvYCgA\n0QwcAvYMzBBmfMMpBmGwIEONUZFyA5gJAEICFuq4U3XkAAAAAElFTkSuQmCC\n",
       "text": [
        " 2\n",
        "x "
       ]
      },
      {
       "latex": [
        "$$x^{2} - y \\left(2^{x + 3} + z\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAAcBAMAAACE+tDfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njaF8d4Zy0YdtD2kc3vYf1SqGVaR9gp0AAAAASUVORK5CYII=\n",
       "text": [
        " 2     \u239b x + 3    \u239e\n",
        "x  - y\u22c5\u239d2      + z\u23a0"
       ]
      }
     ],
     "prompt_number": 46
    }
   ],
   "metadata": {}
  }
 ]
}