Squares and square roots

A perfect square is a number that results from multiplying a whole number by itself: $n^2=n\times n$. We need to check which option equals $n\times n$ for some whole number $n$.

Check: $\sqrt{50}\approx 7.07$ (not whole), $\sqrt{72}\approx 8.49$ (not whole), $\sqrt{81}=9$ because $9\times 9=81$ ✓, $\sqrt{90}\approx 9.49$ (not whole).

Since $9\times 9=81$, the number 81 is a perfect square. It is $9^2$, which appears in the standard list of the first ten perfect squares.

We need to find which option is a perfect square — a number whose square root is a whole number.

50: $7^2=49$ and $8^2=64$, so 50 is between two perfect squares — it is not one itself. 72: $8^2=64$ and $9^2=81$, so 72 is also between two perfect squares — eliminated.

90: $9^2=81$ and $10^2=100$, so 90 falls between two perfect squares and is not a perfect square itself — eliminated.

81 remains. We verify: $9\times 9=81$ ✓. So 81 is the perfect square.

$12^2$ means $12\times 12$. The exponent 2 tells us to multiply the base by itself.

$12\times 12=144$ We can compute this as $12\times 10+12\times 2=120+24=144$.

$12^2=144$.

We need to calculate $12^2=12\times 12$ and match it to one of the options.

24 would be the result of $12\times 2$, which is just doubling — not squaring. Eliminated.

124 and 148 are not the product of $12\times 12$. A quick check: $10\times 12=120$ and $2\times 12=24$, giving $120+24=144$, so neither 124 nor 148 is correct.

144 is the only remaining option, and it matches our calculation $12\times 12=144$ ✓.