Properties of squares and cubes

A perfect square is a whole number multiplied by itself. We need to find which option equals $n\times n$ for some whole number $n$.

Try to find a whole number whose square equals each option: $\sqrt{50}\approx 7.07$ (not whole), $\sqrt{72}\approx 8.49$ (not whole), $\sqrt{81}=9$ (whole!), $\sqrt{90}\approx 9.49$ (not whole).

Since $9\times 9=81$, we have $9^2=81$. Therefore 81 is a perfect square.

We need a number that equals $n^2$ for some whole number $n$. Knowing the list of perfect squares — $1,4,9,16,25,36,49,64,81,100$ — is helpful here.

50 falls between $7^2=49$ and $8^2=64$, so it is not a perfect square. Eliminated.

72 falls between $8^2=64$ and $9^2=81$, so it is not a perfect square. Eliminated.

90 falls between $9^2=81$ and $10^2=100$, so it is not a perfect square. Eliminated.

81 remains. Indeed, $9^2=9\times 9=81$, confirming it is a perfect square.

$\sqrt{144}$ asks: "What number multiplied by itself gives 144?" We need to find $n$ such that $n\times n=144$.

Try $n=12$: $12\times 12=144$. This works!

Since $12^2=144$, we have $\sqrt{144}=12$.

We need to find which option, when squared, equals 144. Square each option and compare.

$11^2=11\times 11=121\ne 144$. 11 is eliminated.

$13^2=13\times 13=169\ne 144$. 13 is eliminated.

$14^2=14\times 14=196\ne 144$. 14 is eliminated.

Only 12 remains. Verify: $12\times 12=144$ ✓, so $\sqrt{144}=12$.