Question

learning goal from 11.2
i can rewrite expressions that contain radicals and/or rational
exponents using the properties of exponents.
lesson reflection (circle one)
starting...
getting there...
got it!
write the equivalent expression for the following radical expressions. simplify numerical expressions
when possible. (1 point)

1. ((sqrt{x})^{9})
2. (sqrt3{7^{14}})

simplify each expression. assume all variables are positive. (1 point)

1. (\frac{27^{\frac{3}{4}}}{27^{\frac{1}{2}}})
2. (3sqrt3{81x^{4}y^{7}})
3. (\frac{sqrt3{27}}{sqrt{8}})

Explanation:

Step1: Use the rule \((\sqrt[n]{a})^m=a^{\frac{m}{n}}\)

For \(10.(\sqrt{x})^9\), here \(n = 2\), \(m=9\), so \((\sqrt{x})^9=x^{\frac{9}{2}}\)

Step2: Use the rule \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\)

For \(11.\sqrt[3]{7^{14}}\), here \(n = 3\), \(m = 14\), so \(\sqrt[3]{7^{14}}=7^{\frac{14}{3}}\)

Step3: Use the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)

For \(8.x^{\frac{3}{2}}\), it is equivalent to \(\sqrt{x^{3}}\)
For \(9.27^{\frac{4}{3}}\), first \(27 = 3^3\), then \(27^{\frac{4}{3}}=(3^3)^{\frac{4}{3}}\)

Step4: Use the rule \((a^m)^n=a^{mn}\)

\((3^3)^{\frac{4}{3}}=3^{3\times\frac{4}{3}}=3^4 = 81\)
For \(12.27^{\frac{1}{2}}\div27^{\frac{3}{4}}\), use the rule \(a^m\div a^n=a^{m - n}\)
\(27^{\frac{1}{2}-\frac{3}{4}}=27^{-\frac{1}{4}}=\frac{1}{27^{\frac{1}{4}}}\)
For \(13.3\sqrt[3]{81x^{4}y^{2}}\), \(81=3^4\), so \(\sqrt[3]{81x^{4}y^{2}}=\sqrt[3]{3^4x^{4}y^{2}}=3x\sqrt[3]{3xy^{2}}\), then \(3\sqrt[3]{81x^{4}y^{2}}=9x\sqrt[3]{3xy^{2}}\)
For \(14.\frac{\sqrt[3]{27}}{\sqrt{8}}\), \(\sqrt[3]{27}=3\), \(\sqrt{8}=2\sqrt{2}\), so \(\frac{\sqrt[3]{27}}{\sqrt{8}}=\frac{3}{2\sqrt{2}}=\frac{3\sqrt{2}}{4}\)

Answer:

1. \(\sqrt{x^{3}}\)
2. \(81\)
3. \(x^{\frac{9}{2}}\)
4. \(7^{\frac{14}{3}}\)
5. \(\frac{1}{27^{\frac{1}{4}}}\)
6. \(9x\sqrt[3]{3xy^{2}}\)
7. \(\frac{3\sqrt{2}}{4}\)