Exponents and radicals — Topic practice

x⁷, with x ≠ 0

First apply the power-of-a-power rule, then the quotient rule:

$\frac{\text{(}{\text{x}}^{\text{4}}{\text{)}}^{\text{3}}}{{\text{x}}^{\text{5}}}$ = $\frac{{\text{x}}^{\text{12}}}{{\text{x}}^{\text{5}}}$ = x¹²⁻⁵ = x⁷.

The original fraction excludes zero. Multiplying 4 and 3 is correct for a power raised to a power; subtracting 5 is correct for division of powers with the same nonzero base. Review 2.01 and 2.02.

6$\sqrt{\text{3}}$

Extract the largest perfect-square factors:

$\sqrt{\text{48}}$ + $\sqrt{\text{12}}$ = $\sqrt{\text{16 ⋅ 3}}$ + $\sqrt{\text{4 ⋅ 3}}$ = 4$\sqrt{\text{3}}$ + 2$\sqrt{\text{3}}$ = 6$\sqrt{\text{3}}$.

Only like radicals combine by adding coefficients. Replacing the sum with $\sqrt{\text{60}}$ would incorrectly distribute a square root over addition. Review 2.06.

3|z|

The principal square root is nonnegative. Therefore

$\sqrt{{\text{9z}}^{\text{2}}}$ = $\sqrt{\text{9}}$$\sqrt{{\text{z}}^{\text{2}}}$ = 3|z|.

Using 3z fails when z < 0: for example, z = −2 gives a square root of 6, not −6. The answer also works at z = 0. Review 2.07.