거듭제곱근과 지수의 확장

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ ${({r}^{16}{s}^{10})}^{\frac{1}{2}}$ ⓑ ${({u}^{10}{v}^{5})}^{\frac{4}{5}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{{r}^{\frac{5}{2}}\cdot {r}^{-\frac{1}{2}}}{{r}^{-\frac{3}{2}}}$ ⓑ $\frac{{s}^{\frac{1}{5}}\cdot s}{{s}^{-\frac{9}{5}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{{a}^{\frac{3}{4}}\cdot {a}^{-\frac{1}{4}}}{{a}^{-\frac{10}{4}}}$ ⓑ $\frac{{b}^{\frac{2}{3}}\cdot b}{{b}^{-\frac{7}{3}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{{c}^{\frac{5}{3}}\cdot {c}^{-\frac{1}{3}}}{{c}^{-\frac{2}{3}}}$ ⓑ $\frac{{d}^{\frac{3}{5}}\cdot d}{{d}^{-\frac{2}{5}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{{m}^{\frac{7}{4}}\cdot {m}^{-\frac{5}{4}}}{{m}^{-\frac{2}{4}}}$ ⓑ $\frac{{n}^{\frac{3}{7}}\cdot n}{{n}^{-\frac{4}{7}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${4}^{\frac{5}{2}}\cdot {4}^{\frac{1}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${n}^{\frac{2}{6}}\cdot {n}^{\frac{4}{6}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${({a}^{24})}^{\frac{1}{6}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${({b}^{10})}^{\frac{3}{5}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

$\frac{{w}^{\frac{2}{5}}}{{w}^{\frac{7}{5}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

$\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{8}{3}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${\left(27{r}^{\frac{3}{5}}\right)}^{\frac{1}{3}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${\left(64{s}^{\frac{3}{5}}\right)}^{\frac{1}{6}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${({r}^{9}{s}^{12})}^{\frac{1}{3}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${({u}^{12}{v}^{18})}^{\frac{1}{6}}$

Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify ${144}^{\frac{1}{2}}$ to find the length of each side of his garden.

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify ${242}^{\frac{1}{2}}$ to find the length of each side of his patio.Round to the nearest tenth of a foot.

Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify $\frac{{12}^{\frac{1}{2}}}{{16}^{\frac{1}{2}}}$ to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify $\frac{{1024}^{\frac{1}{2}}}{{16}^{\frac{1}{2}}}$ to find how many seconds it took for the flare to reach the water.

Show two different algebraic methods to simplify ${4}^{\frac{3}{2}}.$ Explain all your steps.