Fractional Exponents and Radicals: Practice

The product rule forces $a^{1/2}\cdot a^{1/2}=a^{1}=a$, so $a^{1/2}$ is the number that squares to $a$.

$a^{1/2}=\sqrt{a}$

That number is the square root of $a$, not $2a$ and not $\tfrac{a}{2}$.

The cube root asks for the number whose cube is $8$.

$\sqrt[3]{8}=2$

Check: $2^3=8$.

A one-over-three power is the cube root, the number whose cube is $27$.

$27^{1/3}=\sqrt[3]{27}=3$

Check: $3^3=27$.

A root is a one-over-index power, and a square root has index $2$.

$\sqrt{a}=a^{1/2}$

The cube root asks what number cubed gives $1$.

$\sqrt[3]{1}=1$

Check: $1^3=1$.

The exponent $\tfrac{1}{3}$ is the cube root.

$8^{1/3}=\sqrt[3]{8}=2$

Check: $2^3=8$.

Find the number whose square is $36$.

$\sqrt{36}=6$

Check: $6^2=36$.

The index is the small number that names the order of the root; the number underneath is the radicand.

$\sqrt[3]{10}:\ \text{index }3,\ \text{radicand }10$

A square root has index $2$, which becomes the denominator of the exponent.

$\sqrt{7}=7^{1/2}$

The fifth root asks for the number whose fifth power is $32$.

$\sqrt[5]{32}=2$

Check: $2^5=32$.

Find the number whose square is $64$.

$\sqrt{64}=8$

Check: $8^2=64$.

Find the number whose square is $49$.

$\sqrt{49}=7$

Check: $7^2=49$.