Question

write the equivalent expression for the following expressions with rational exponents. simplify numerical expressions when possible. (1 point each)

1. $x^{\frac{3}{2}}$ 9. $27^{\frac{2}{3}}

write the equivalent expression for the following radical expressions. simplify numerical expressions when possible. (1 point)

1. $(sqrt{x})^{9}$ 11. $sqrt7{7^{14}}

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

1. $\frac{27^{\frac{4}{3}}}{27^{\frac{2}{3}}}$ 13. $3sqrt3{81x^{4}y^{2}}$ 14. $sqrt3{\frac{27}{8}}$

Explanation:

Step1: Recall exponent - radical conversion rule

The rule is $a^{\frac{m}{n}}=\sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}$. For $x^{\frac{3}{2}}$, by the rule, it is equivalent to $\sqrt{x^{3}}=(\sqrt{x})^{3}$.

Step2: Simplify $27^{\frac{2}{3}}$

First, rewrite $27$ as $3^{3}$. Then $27^{\frac{2}{3}}=(3^{3})^{\frac{2}{3}}$. According to the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, we have $(3^{3})^{\frac{2}{3}} = 3^{3\times\frac{2}{3}}=3^{2}=9$.

Step3: Simplify $(\sqrt{x})^{9}$

Using the conversion $(\sqrt{x})^{9}=x^{\frac{9}{2}}$.

Step4: Simplify $\sqrt[7]{7^{14}}$

By the rule $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$, we get $\sqrt[7]{7^{14}}=7^{\frac{14}{7}} = 7^{2}=49$.

Step5: Simplify $\frac{27^{\frac{4}{3}}}{27^{\frac{2}{3}}}$

According to the rule $\frac{a^{m}}{a^{n}}=a^{m - n}$, we have $27^{\frac{4}{3}-\frac{2}{3}}=27^{\frac{2}{3}}$. Since $27 = 3^{3}$, then $27^{\frac{2}{3}}=(3^{3})^{\frac{2}{3}}=3^{2}=9$.

Step6: Simplify $3\sqrt[3]{81x^{4}y^{2}}$

Rewrite $81$ as $3^{4}$, then $3\sqrt[3]{81x^{4}y^{2}}=3\sqrt[3]{3^{4}x^{4}y^{2}}=3\times3^{\frac{4}{3}}x^{\frac{4}{3}}y^{\frac{2}{3}}$. Using the rule $a^{m}\times a^{n}=a^{m + n}$, we get $3^{1+\frac{4}{3}}x^{\frac{4}{3}}y^{\frac{2}{3}}=3^{\frac{3 + 4}{3}}x^{\frac{4}{3}}y^{\frac{2}{3}}=3^{\frac{7}{3}}x^{\frac{4}{3}}y^{\frac{2}{3}}=\ 3^{2+\frac{1}{3}}x^{\frac{4}{3}}y^{\frac{2}{3}}=9\sqrt[3]{3x^{4}y^{2}}$.

Step7: Simplify $\sqrt[3]{\frac{27}{8}}$

We know that $27 = 3^{3}$ and $8 = 2^{3}$, so $\sqrt[3]{\frac{27}{8}}=\frac{\sqrt[3]{27}}{\sqrt[3]{8}}=\frac{3}{2}$.

Answer:

1. $(\sqrt{x})^{3}$ or $\sqrt{x^{3}}$
2. $9$
3. $x^{\frac{9}{2}}$
4. $49$
5. $9$
6. $9\sqrt[3]{3x^{4}y^{2}}$
7. $\frac{3}{2}$