Question

1 other rational exponents
rewrite each expression using one or more roots (like $sqrt{2}$ or $sqrt3{2}$) and whole - number exponents.
1 $left(11^{\frac{1}{2}}
ight)^3$
2 $left(b^5
ight)^{\frac{1}{4}}$
3 $3^{\frac{2}{5}}$
4 $left(a^2cdot b^4
ight)^{\frac{5}{3}}$
5 $left(x^3cdot b^9
ight)^{\frac{2}{3}}$
6 $left(y^{\frac{1}{2}}z^{\frac{5}{4}}
ight)^8$

Explanation:

Step1: Apply power of a power rule

For \((11^{\frac{1}{2}})^3\): use \((a^m)^n=a^{m\cdot n}\)
\(11^{\frac{1}{2}\cdot3}=11^{\frac{3}{2}}=11^{1+\frac{1}{2}}=11\cdot\sqrt{11}\)

Step2: Apply power of a power rule

For \((b^3)^{\frac{1}{4}}\): use \((a^m)^n=a^{m\cdot n}\)
\(b^{3\cdot\frac{1}{4}}=b^{\frac{3}{4}}=\sqrt[4]{b^3}\)

Step3: Rewrite rational exponent as root

For \(3^{\frac{2}{5}}\): use \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)
\(\sqrt[5]{3^2}=\sqrt[5]{9}\)

Step4: Apply power of product/rule

For \((a^2\cdot b^4)^{\frac{1}{3}}\): use \((ab)^n=a^n b^n\) and \((a^m)^n=a^{m\cdot n}\)
\((a^2)^{\frac{1}{3}}\cdot(b^4)^{\frac{1}{3}}=a^{\frac{2}{3}}\cdot b^{\frac{4}{3}}=\sqrt[3]{a^2}\cdot b\sqrt[3]{b}\)

Step5: Apply power of product/rule

For \((x^3\cdot b^9)^{\frac{2}{3}}\): use \((ab)^n=a^n b^n\) and \((a^m)^n=a^{m\cdot n}\)
\((x^3)^{\frac{2}{3}}\cdot(b^9)^{\frac{2}{3}}=x^{2}\cdot b^{6}\)

Step6: Apply power of product/rule

For \((y^{\frac{1}{2}} z^{\frac{3}{4}})^8\): use \((ab)^n=a^n b^n\) and \((a^m)^n=a^{m\cdot n}\)
\((y^{\frac{1}{2}})^8\cdot(z^{\frac{3}{4}})^8=y^{4}\cdot z^{6}\)

Answer:

1. \(11\sqrt{11}\)
2. \(\sqrt[4]{b^3}\)
3. \(\sqrt[5]{9}\)
4. \(b\sqrt[3]{a^2 b}\)
5. \(x^2 b^6\)
6. \(y^4 z^6\)