Question

1. circle all of the expressions that are equivalent to $sqrt5{32x^{3}y^{-10}z^{12}}$.

a. $6.4y^{2}z^{2}sqrt5{x^{3}z^{2}}$
b. $\frac{2x^{2}sqrt5{x^{3}z^{2}}}{y^{2}}$
c. $\frac{2x^{\frac{3}{5}}z^{\frac{12}{5}}}{y^{2}}$
d. $2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}$
e. $2y^{2}z^{2}sqrt5{x^{3}z^{2}}$

Explanation:

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

$$
LATEXBLOCK0
$$

Step2: Use the property \((ab)^n=a^nb^n\)

$$
LATEXBLOCK1
$$

Step3: Simplify \(32^{\frac{1}{5}}\)

Since \(32 = 2^5\), then \(32^{\frac{1}{5}}=(2^5)^{\frac{1}{5}} = 2\)

$$
LATEXBLOCK2
$$

Step4: Rewrite \(y^{-2}\) as \(\frac{1}{y^{2}}\)

$$
LATEXBLOCK3
$$

Step5: Check each option

• Option a: \(6.4y^{2}z^{2}\sqrt[5]{x^{3}z^{2}}\)

\(\sqrt[5]{x^{3}z^{2}}=x^{\frac{3}{5}}z^{\frac{2}{5}}\), and \(6.4y^{2}z^{2}\sqrt[5]{x^{3}z^{2}}=6.4y^{2}z^{2 +\frac{2}{5}}x^{\frac{3}{5}}
eq2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}\)

• Option b: \(\frac{2x^{\frac{3}{5}}z^{\frac{12}{5}}}{y^{2}}\)

This is equivalent to our simplified form \(2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}\)

• Option c: \(\frac{2x^{\frac{3}{5}}z^{\frac{12}{5}}}{y^{2}}\)

This is equivalent to our simplified form \(2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}\)

• Option d: \(2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}\)

This is our simplified form

• Option e: \(2y^{2}z^{2}\sqrt[5]{x^{3}z^{2}}\)

\(\sqrt[5]{x^{3}z^{2}}=x^{\frac{3}{5}}z^{\frac{2}{5}}\), and \(2y^{2}z^{2}\sqrt[5]{x^{3}z^{2}}=2y^{2}z^{2+\frac{2}{5}}x^{\frac{3}{5}}
eq2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}\)

Answer:

B. \(\frac{2x^{\frac{3}{5}}\sqrt[5]{x^{3}z^{2}}}{y^{2}}\), C. \(\frac{2x^{\frac{3}{5}}z^{\frac{12}{5}}}{y^{2}}\), D. \(2x^{\frac{3}{5}}y^{-2}z^{\frac{12}{5}}\)