Expressions and Equations: Working with Square Roots and Cube Roots (CCSS.8.EE.2)

The square root of a number is the nonnegative value that multiplied by itself gives the original number. Since $9 \times 9 = 81$, $\sqrt{81} = 9$. $81$ is the radicand (not the root), $-9$ is negative (principal square roots are nonnegative), and $3$ is too small (e.g., $3^2 = 9$).

The cube root of a number is the value that multiplied by itself three times gives the original number. Since $4 \times 4 \times 4 = 64$, $\sqrt[3]{64} = 4$. $8$ is the square root of $64$, $-4$ is negative (but the cube root of a positive is positive), and $64$ is the radicand.

The square root is the nonnegative number whose square equals the original number. Since $12 \times 12 = 144$, $\sqrt{144} = 12$. $144$ is the radicand, $-12$ is negative (principal square roots are nonnegative), and $6$ is the square root of $36$, not $144$.

The cube root is the number that multiplied by itself three times gives the original number. Since $5 \times 5 \times 5 = 125$, $\sqrt[3]{125} = 5$. $125$ is the radicand, $-5$ is negative (the cube root of a positive is positive), and $25$ confuses the operation (it is not a cube root here).

The square root of 81 is the nonnegative number whose square is 81, which is 9. 81 is the radicand, -9 is negative (the principal square root is nonnegative), and 8 confuses square and cube roots.

The cube root of 125 is the number that multiplied by itself three times equals 125, which is 5. 125 is the radicand, -5 is negative (but $(-5)^3=-125$), and 25 confuses the result by squaring 5.

The square root of 144 is the nonnegative number whose square is 144, which is 12. 144 is the radicand, -12 is negative (not the principal square root), and 14 is an arithmetic mistake.

The cube root of 27 is the number that multiplied by itself three times equals 27, which is 3. 9 confuses with the square root of 81, -3 is negative ($(-3)^3=-27$), and 27 is the radicand.

$\sqrt{81}$ is the positive number whose square is 81, so it equals 9 because $9^2=81$. 81 is the radicand, not the root; $-9$ is negative (not the principal square root); 3 is incorrect since $3^2=9$, not 81.

Solving $x^2=121$ gives $x=\pm 11$. The positive solution (and the value of $\sqrt{121}$) is 11 because $11^2=121$. 121 is the radicand, and $-11$ is the negative solution.