How do you simplify #((3m^(1/6) n^(1/3)) / (4n^(-2/3))) ^2#?

Question

Eva Adamson · Accepted Answer

See the entire simplification process below:

First, we will use this rule of exponents to simplify the term within the parenthesis: #x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))# #((3m^(1/6)n^(color(red)(1/3)))/(4n^(color(blue)(-2/3))))^2 = ((3m^(1/6)n^(color(red)(1/3)-color(blue)(-2/3)))/4)^2 = ((3m^(1/6)n^(3/3))/4)^2 =# #((3m^(1/6)n^1)/4)^2# Now, use these two rules for exponents to simplify the terms for the entire expression: #a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))# #((3^color(red)(1)m^(color(red)(1/6))n^color(red)(1))/4^color(red)(1))^color(blue)(2) = (3^(color(red)(1)xxcolor(blue)(2))m^(color(red)(1/6)xxcolor(blue)(2))n^(color(red)(1)xxcolor(blue)(2)))/4^(color(red)(1)xxcolor(blue)(2)) = (3^2m^(2/6)n^2)/4^2 = (9m^(1/3)n^2)/16#

Abigail Allen · Answer

undefined To simplify the expression $((3m^\frac{1}{6} n^\frac{1}{3}) / (4n^{-\frac{2}{3}}))^2$:

First, let's simplify the numerator and the denominator separately:

Numerator:
$3m^\frac{1}{6} n^\frac{1}{3}$

Denominator:
$4n^{-\frac{2}{3}}$

Now, let's simplify each term:

Numerator:
$3m^\frac{1}{6} n^\frac{1}{3} = \frac{3}{1} \cdot m^\frac{1}{6} \cdot n^\frac{1}{3} = 3m^\frac{1}{6} n^\frac{1}{3}$

Denominator:
$4n^{-\frac{2}{3}} = \frac{4}{1} \cdot n^{-\frac{2}{3}} = \frac{4}{n^\frac{2}{3}}$

Now, let's rewrite the expression with the simplified terms:

$((3m^\frac{1}{6} n^\frac{1}{3}) / (4n^{-\frac{2}{3}}))^2 = \left(\frac{3m^\frac{1}{6} n^\frac{1}{3}}{\frac{4}{n^\frac{2}{3}}}\right)^2$

Now, let's simplify the expression inside the parentheses:

$\frac{3m^\frac{1}{6} n^\frac{1}{3}}{\frac{4}{n^\frac{2}{3}}} = \frac{3m^\frac{1}{6} n^\frac{1}{3} \cdot n^\frac{2}{3}}{4}$

Using the properties of exponents, we can combine the terms:

$= \frac{3m^\frac{1}{6} n^{\frac{1}{3} + \frac{2}{3}}}{4} = \frac{3m^\frac{1}{6} n}{4}$

Now, let's square this expression:

$\left(\frac{3m^\frac{1}{6} n}{4}\right)^2 = \left(\frac{3m^\frac{1}{6} n}{4}\right) \cdot \left(\frac{3m^\frac{1}{6} n}{4}\right)$

Using the properties of exponents and multiplication of fractions, we have:

$= \frac{(3m^\frac{1}{6})^2 n^2}{4^2} = \frac{9m^\frac{1}{3} n^2}{16}$

Therefore, the simplified expression is $\frac{9m^\frac{1}{3} n^2}{16}$.