Question

1. reinforce simplify the following expressions.

a. $(16x^{3})^{\frac {1}{4}}=$
b. $\frac {4}{(8x^{3})^{\frac {1}{3}}}=$
c. $\frac {(8x^{3})^{\frac {1}{3}}}{(4x^{2})^{\frac {1}{2}}}=$
d. $(2x^{2}y)^{3}\cdot (x^{3}y^{4}z^{-1})^{\frac {1}{3}}=$

Explanation:

Step1: Simplify \( (16x^{3})^{\frac{1}{4}} \)

Use the power - of - a - product rule \((ab)^n=a^nb^n\). So, \((16x^{3})^{\frac{1}{4}} = 16^{\frac{1}{4}}\cdot(x^{3})^{\frac{1}{4}}\).
Since \(16 = 2^{4}\), then \(16^{\frac{1}{4}}=(2^{4})^{\frac{1}{4}} = 2\). And \((x^{3})^{\frac{1}{4}}=x^{\frac{3}{4}}\).
So, \((16x^{3})^{\frac{1}{4}}=2x^{\frac{3}{4}}\).

Step2: Simplify \(\frac{4}{(8x^{3})^{\frac{1}{3}}}\)

First, simplify \((8x^{3})^{\frac{1}{3}}\). Using \((ab)^n=a^nb^n\), we have \((8x^{3})^{\frac{1}{3}}=8^{\frac{1}{3}}\cdot(x^{3})^{\frac{1}{3}}\).
Since \(8 = 2^{3}\), then \(8^{\frac{1}{3}}=(2^{3})^{\frac{1}{3}} = 2\). And \((x^{3})^{\frac{1}{3}}=x\).
So, \(\frac{4}{(8x^{3})^{\frac{1}{3}}}=\frac{4}{2x}=\frac{2}{x}\).

Step3: Simplify \(\frac{(8x^{3})^{\frac{1}{3}}}{(4x^{2})^{\frac{1}{2}}}\)

Simplify numerator and denominator separately.
For the numerator: \((8x^{3})^{\frac{1}{3}}=8^{\frac{1}{3}}\cdot(x^{3})^{\frac{1}{3}} = 2x\).
For the denominator: \((4x^{2})^{\frac{1}{2}}=4^{\frac{1}{2}}\cdot(x^{2})^{\frac{1}{2}}=2x\).
Then \(\frac{(8x^{3})^{\frac{1}{3}}}{(4x^{2})^{\frac{1}{2}}}=\frac{2x}{2x}=1\).

Step4: Simplify \((2x^{2}y)^{3}\cdot(x^{3}y^{4}z^{-1})^{\frac{1}{3}}\)

First, expand \((2x^{2}y)^{3}\) using \((ab)^n=a^nb^n\): \((2x^{2}y)^{3}=2^{3}\cdot(x^{2})^{3}\cdot y^{3}=8x^{6}y^{3}\).
Then expand \((x^{3}y^{4}z^{-1})^{\frac{1}{3}}\) using \((ab)^n=a^nb^n\): \((x^{3}y^{4}z^{-1})^{\frac{1}{3}}=x^{3\times\frac{1}{3}}y^{4\times\frac{1}{3}}z^{-1\times\frac{1}{3}}=x y^{\frac{4}{3}}z^{-\frac{1}{3}}\).
Multiply the two results: \(8x^{6}y^{3}\cdot x y^{\frac{4}{3}}z^{-\frac{1}{3}}=8x^{6 + 1}y^{3+\frac{4}{3}}z^{-\frac{1}{3}}=8x^{7}y^{\frac{9 + 4}{3}}z^{-\frac{1}{3}}=8x^{7}y^{\frac{13}{3}}z^{-\frac{1}{3}}=\frac{8x^{7}y^{\frac{13}{3}}}{z^{\frac{1}{3}}}\).

Answer:

a. \(2x^{\frac{3}{4}}\)
b. \(\frac{2}{x}\)
c. \(1\)
d. \(\frac{8x^{7}y^{\frac{13}{3}}}{z^{\frac{1}{3}}}\)