How do you evaluate and simplify #(120^(-2/5)*120^(2/5))/7^(-3/4)#?

Question

Hazel Anglin · Accepted Answer

See a solution process below:

First, use these rules of exponents to simplify the numerator: #x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))# and #a^color(red)(0) = 1# #(120^color(red)(-2/5) * 120^color(blue)(2/5))/7^(-3/4) => 120^(color(red)(-2/5)+color(blue)(2/5))/7^(-3/4) =># #120^color(red)(0)/7^(-3/4) => 1/7^(-3/4)# Next, we will use this rule to rewrite the expression: #1/x^color(red)(a) = x^color(red)(-a)# #1/7^color(red)(-3/4) = 7^color(red)(- -3/4) = 7^(3/4)# Then, we can rewrite the expression as: #7^(3 xx 1/4)# Now, we can use this rule of exponents to continue the simplification: #x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)# #7^(color(red)(3) xx color(blue)(1/4)) => (7^color(red)(3))^color(blue)(1/4) => 343^(1/4)# Or, using this rule: #x^(1/color(red)(n)) = root(color(red)(n))(x)# #root(4)(343)#

Eva Adamson · Answer

undefined To evaluate and simplify $ \left(120^{-2/5} \times 120^{2/5}\right) \div 7^{-3/4} $, you can follow these steps:

1. Use the properties of exponents to combine the terms inside the parentheses.
2. Simplify each term raised to a power.
3. Combine the terms and simplify further if possible.

Let's break it down:

$$ \left(120^{-2/5} \times 120^{2/5}\right) \div 7^{-3/4} $$

$$ = 120^{(-2/5 + 2/5)} \div 7^{-3/4} $$

$$ = 120^{0} \div 7^{-3/4} $$

$$ = 1 \div (7^{-3/4}) $$

$$ = 7^{3/4} $$

$$ = \sqrt[4]{7^3} $$

$$ = \sqrt[4]{343} $$

So, $ \left(120^{-2/5} \times 120^{2/5}\right) \div 7^{-3/4} $ simplifies to $ \sqrt[4]{343} $.