Answered

Describe how you would simplify the given expression.

[tex]\[ \left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}, \quad x \neq 0, \quad y \neq 0 \][/tex]

[tex]\[\square\][/tex]



Answer :

Certainly! Let's simplify the expression step by step.

Given:
[tex]\[ \left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3} \][/tex]

1. Simplify inside the parentheses:

- Simplify the coefficients:
[tex]\[ \frac{20}{5} = 4 \][/tex]

- Apply the rules of exponents for the [tex]\(x\)[/tex] terms:
[tex]\[ \frac{x^5}{x^{-3}} = x^{5 - (-3)} = x^{5 + 3} = x^8 \][/tex]

- Apply the rules of exponents for the [tex]\(y\)[/tex] terms:
[tex]\[ \frac{y^2}{y^7} = y^{2 - 7} = y^{-5} \][/tex]

- Combine these simplified pieces:
[tex]\[ \frac{20 x^5 y^2}{5 x^{-3} y^7} = 4 x^8 y^{-5} \][/tex]

2. Apply the outer exponent of [tex]\(-3\)[/tex]:

- For the coefficient [tex]\(4\)[/tex]:
[tex]\[ 4^{-3} = \left(\frac{1}{4}\right)^3 = \frac{1}{4^3} = \frac{1}{64} = 0.015625 \][/tex]

- For the [tex]\(x\)[/tex] term:
[tex]\[ (x^8)^{-3} = x^{8 \cdot (-3)} = x^{-24} \][/tex]

- For the [tex]\(y\)[/tex] term:
[tex]\[ (y^{-5})^{-3} = y^{-5 \cdot (-3)} = y^{15} \][/tex]

3. Combine the results:

[tex]\[ \left(4 x^8 y^{-5}\right)^{-3} = 0.015625 x^{-24} y^{15} \][/tex]

Therefore, the final simplified expression is:

[tex]\[ 0.015625 x^{-24} y^{15} \][/tex]