거듭제곱근과 지수의 확장

Simplify Expressions with ${a}^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[3]{a}$ ⓑ ${\left(\sqrt[4]{21v}\right)}^{3}$ ⓒ $\sqrt[4]{{\left(\frac{2xy}{5z}\right)}^{2}}$

In the following exercises, simplify.

ⓐ ${64}^{\frac{5}{2}}$ ⓑ ${81}^{\frac{-3}{2}}$ ⓒ ${(-27)}^{\frac{2}{3}}$

In the following exercises, simplify.

ⓐ ${25}^{\frac{3}{2}}$ ⓑ ${9}^{-\frac{3}{2}}$ ⓒ ${(-64)}^{\frac{2}{3}}$

In the following exercises, simplify.

ⓐ ${32}^{\frac{2}{5}}$ ⓑ ${27}^{-\frac{2}{3}}$ ⓒ ${(-25)}^{\frac{1}{2}}$

In the following exercises, simplify.

ⓐ ${100}^{\frac{3}{2}}$ ⓑ ${49}^{-\frac{5}{2}}$ ⓒ ${(-100)}^{\frac{3}{2}}$

In the following exercises, simplify.

ⓐ $-{9}^{\frac{3}{2}}$ ⓑ $-{9}^{-\frac{3}{2}}$ ⓒ ${(-9)}^{\frac{3}{2}}$

In the following exercises, simplify.

ⓐ $-{64}^{\frac{3}{2}}$ ⓑ $-{64}^{-\frac{3}{2}}$ ⓒ ${(-64)}^{\frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${c}^{\frac{1}{4}}\cdot {c}^{\frac{5}{8}}$ ⓑ ${({p}^{12})}^{\frac{3}{4}}$ ⓒ $\frac{{r}^{\frac{4}{5}}}{{r}^{\frac{9}{5}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${6}^{\frac{5}{2}}\cdot {6}^{\frac{1}{2}}$ ⓑ ${({b}^{15})}^{\frac{3}{5}}$ ⓒ $\frac{{w}^{\frac{2}{7}}}{{w}^{\frac{9}{7}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${y}^{\frac{1}{2}}\cdot {y}^{\frac{3}{4}}$ ⓑ ${({x}^{12})}^{\frac{2}{3}}$ ⓒ $\frac{{m}^{\frac{5}{8}}}{{m}^{\frac{13}{8}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${q}^{\frac{2}{3}}\cdot {q}^{\frac{5}{6}}$ ⓑ ${({h}^{6})}^{\frac{4}{3}}$ ⓒ $\frac{{n}^{\frac{3}{5}}}{{n}^{\frac{8}{5}}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${\left(27{q}^{\frac{3}{2}}\right)}^{\frac{4}{3}}$ ⓑ ${\left({a}^{\frac{1}{3}}{b}^{\frac{2}{3}}\right)}^{\frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${\left(64{s}^{\frac{3}{7}}\right)}^{\frac{1}{6}}$ ⓑ ${\left({m}^{\frac{4}{3}}{n}^{\frac{1}{2}}\right)}^{\frac{3}{4}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${\left(16\,{u}^{\frac{1}{3}}\right)}^{\frac{3}{4}}$ ⓑ ${\left(4\,{p}^{\frac{1}{3}}{q}^{\frac{1}{2}}\right)}^{\frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${\left(625\,{n}^{\frac{8}{3}}\right)}^{\frac{3}{4}}$ ⓑ ${\left(9\,{x}^{\frac{2}{5}}{y}^{\frac{3}{5}}\right)}^{\frac{5}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $\frac{{r}^{\frac{5}{2}}\cdot {r}^{-\frac{1}{2}}}{{r}^{-\frac{3}{2}}}$ ⓑ ${\left(\frac{36\,{s}^{\frac{1}{5}}{t}^{-\frac{3}{2}}}{{s}^{-\frac{9}{5}}{t}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $\frac{{a}^{\frac{3}{4}}\cdot {a}^{-\frac{1}{4}}}{{a}^{-\frac{10}{4}}}$ ⓑ ${\left(\frac{27\,{b}^{\frac{2}{3}}{c}^{-\frac{5}{2}}}{{b}^{-\frac{7}{3}}{c}^{\frac{1}{2}}}\right)}^{\frac{1}{3}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $\frac{{c}^{\frac{5}{3}}\cdot {c}^{-\frac{1}{3}}}{{c}^{-\frac{2}{3}}}$ ⓑ ${\left(\frac{8\,{x}^{\frac{5}{3}}\,{y}^{-\frac{1}{2}}}{27\,{x}^{-\frac{4}{3}}\,{y}^{\frac{5}{2}}}\right)}^{\frac{1}{3}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $\frac{{m}^{\frac{7}{4}}\cdot {m}^{-\frac{5}{4}}}{{m}^{-\frac{2}{4}}}$ ⓑ ${\left(\frac{16\,{m}^{\frac{1}{5}}\,{n}^{\frac{3}{2}}}{81\,{m}^{\frac{9}{5}}\,{n}^{-\frac{1}{2}}}\right)}^{\frac{1}{4}}$

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