Try the following questions on square and cube numbers.
", "advice": "Squared integers are called square numbers. It may be useful to remember the first few square numbers to be able to use them without a calculator.
\nHere:
\n\\[ \\begin{align} \\var{x[0]}^2 &= \\var{x[0]} \\times \\var{x[0]} \\\\&= \\var{x[0]^2} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[1]}^2 &= \\var{x[1]} \\times \\var{x[1]} \\\\&= \\var{x[1]^2} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[2]}^2 &= \\var{x[2]} \\times \\var{x[2]} \\\\&= \\var{x[2]^2} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[3]}^2 &= \\var{x[3]} \\times \\var{x[3]} \\\\&= \\var{x[3]^2} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[4]}^2 &= \\var{x[4]} \\times \\var{x[4]} \\\\&= \\var{x[4]^2} \\text{.}\\end{align}\\]
\n\nCubed integers are called cubed numbers. To obtain these, we would typically always use a calculator.
\nWe can either cube the number $x$:
\n\\[ \\begin{align} \\var{x[0]}^3 &= \\var{x[0]} \\times \\var{x[0]} \\times \\var{x[0]} \\\\&= \\var{x[0]^3} \\text{,} \\end{align}\\]
\nor we can multiply the square number $(x_n)^2$ from part a) by the appropriate $x_n$:
\n\\[ \\begin{align} \\var{x[0]}^3 &= \\var{x[0]}^2 \\times \\var{x[0]} \\\\&= \\var{x[0]^2} \\times \\var{x[0]}\\\\&= \\var{x[0]^3} \\text{.} \\end{align}\\]
\n\\[ \\begin{align} \\var{x[1]}^3 &= \\var{x[1]} \\times \\var{x[1]} \\times \\var{x[1]} \\\\ &= \\var{x[1]^3} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[2]}^3 &= \\var{x[2]} \\times \\var{x[2]} \\times \\var{x[2]} \\\\ &= \\var{x[2]^3} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[3]}^3 &= \\var{x[3]} \\times \\var{x[3]} \\times \\var{x[3]} \\\\ &= \\var{x[3]^3} \\text{.}\\end{align}\\]
\n\\[ \\begin{align} \\var{x[3]}^4 &= \\var{x[4]} \\times \\var{x[4]} \\times \\var{x[4]} \\\\ &= \\var{x[4]^3} \\text{.}\\end{align}\\]
\n\nHere is a table of square numbers for integers from $1$ to $15$:
\n| $y$ | \n$1$ | \n$2$ | \n$3$ | \n$4$ | \n$5$ | \n$6$ | \n$7$ | \n$8$ | \n$9$ | \n$10$ | \n$11$ | \n$12$ | \n$13$ | \n$14$ | \n$15$ | \n
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $y^2$ | \n$1$ | \n$4$ | \n$9$ | \n$16$ | \n$25$ | \n$36$ | \n$49$ | \n$64$ | \n$81$ | \n$100$ | \n$121$ | \n$144$ | \n$169$ | \n$196$ | \n$225$ | \n
The only square number between $\\var{ly}$ and $\\var{uy}$ is $\\var{y2}$.
\nTo calculate $y$ we must calculate the square root of $y^2$,
\n\\[ \\sqrt{\\var{y2}} = \\var{y} \\text{.}\\]
\nThis is our integer $y$.
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\n$\\var{x[0]}^2 =$ [[0]]
\n$\\var{x[1]}^2 =$ [[1]]
\n$\\var{x[2]}^2 =$ [[2]]
\n$\\var{x[3]}^2 =$ [[3]]
\n$\\var{x[4]}^2 =$ [[4]]
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\n$\\var{x[0]}^3 =$ [[0]]
\n$\\var{x[1]}^3 =$ [[1]]
\n$\\var{x[2]}^3 =$ [[2]]
\n$\\var{x[3]}^3 =$ [[3]]
\n$\\var{x[4]}^3 =$ [[4]]
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\n$y^2 = $ [[0]]
\n$y = $ [[1]]
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\nFinally, find a square number between two given limits.
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