Question

lesson 11.1 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

1. which of the following statements are true? select three that apply.

a. $x^{\frac{4}{5}}=sqrt5{x^{4}}$
b. $x^{\frac{2}{8}}=sqrt4{x}$
c. $(x^{\frac{1}{3}})^2 = (sqrt3{x})^2$
d. $(sqrt5{x^{4}})^2=x^{\frac{2}{5}}$
write the equivalent expression for the following expression with rational exponents.

1. $x^{\frac{2}{7}}$

write the equivalent expression for the following radical expressions.

1. $sqrt{x^{6}}$

Explanation:

Step1: Recall radical - exponent rule

The rule is $x^{\frac{m}{n}}=\sqrt[n]{x^{m}}$.

Step2: Analyze option A

For $x^{\frac{4}{5}}$, by the rule $x^{\frac{m}{n}}=\sqrt[n]{x^{m}}$, when $m = 4$ and $n = 5$, we have $x^{\frac{4}{5}}=\sqrt[5]{x^{4}}$, so option A is true.

Step3: Analyze option B

For $x^{\frac{1}{8}}$, by the rule $x^{\frac{m}{n}}=\sqrt[n]{x^{m}}$, it should be $x^{\frac{1}{8}}=\sqrt[8]{x}$, not $\sqrt[4]{x}$, so option B is false.

Step4: Analyze option C

For $(x^{\frac{1}{3}})^2$, first $x^{\frac{1}{3}}=\sqrt[3]{x}$, then $(x^{\frac{1}{3}})^2 = (\sqrt[3]{x})^2$, so option C is true.

Step5: Analyze option D

For $(\sqrt[5]{x^{4}})^2$, since $\sqrt[5]{x^{4}}=x^{\frac{4}{5}}$, then $(\sqrt[5]{x^{4}})^2=(x^{\frac{4}{5}})^2=x^{\frac{8}{5}}$, not $x^{\frac{2}{5}}$, so option D is false.

Step6: Rewrite $x^{\frac{2}{7}}$ as a radical

By the rule $x^{\frac{m}{n}}=\sqrt[n]{x^{m}}$, $x^{\frac{2}{7}}=\sqrt[7]{x^{2}}$.

Step7: Rewrite $\sqrt{x^{6}}$ with a rational - exponent

Since $\sqrt{x^{6}}=(x^{6})^{\frac{1}{2}}$, and using the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, we have $(x^{6})^{\frac{1}{2}}=x^{3}$.

Answer:

1. A. $x^{\frac{4}{5}}=\sqrt[5]{x^{4}}$, C. $(x^{\frac{1}{3}})^2 = (\sqrt[3]{x})^2$
2. $\sqrt[7]{x^{2}}$
3. $x^{3}$