Sure! Let's explain and justify the mathematical steps that were performed to simplify the given expression [tex]\(\frac{(4x^5y^2)^3}{xy^9}\)[/tex].
### Step 1:
Apply the exponent to each term inside the parentheses:
- Use the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to distribute the exponent [tex]\(3\)[/tex] to each factor inside the parentheses:
[tex]\[
(4x^5y^2)^3 = 4^3 (x^5)^3 (y^2)^3
\][/tex]
- Calculate each exponentiation:
[tex]\[
4^3 = 64, \quad (x^5)^3 = x^{15}, \quad (y^2)^3 = y^6
\][/tex]
Therefore, we have:
[tex]\[
\frac{4^3(x^5)^3(y^2)^3}{xy^9} = \frac{64x^{15}y^6}{xy^9}
\][/tex]
### Step 2:
Simplify the expression by dividing the numerator by the denominator:
- Use the quotient rule for exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex] to simplify the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
\frac{64x^{15}y^6}{x^1y^9} = 64x^{15-1}y^{6-9}
\][/tex]
Simplify the exponents:
[tex]\[
x^{15-1} = x^{14}, \quad y^{6-9} = y^{-3}
\][/tex]
So we obtain:
[tex]\[
64x^{14}y^{-3}
\][/tex]
### Step 3:
Express negative exponents as positive by rewriting [tex]\(y^{-3}\)[/tex]:
- Recall that [tex]\(a^{-m} = \frac{1}{a^m}\)[/tex]:
[tex]\[
y^{-3} = \frac{1}{y^3}
\][/tex]
Therefore, the expression [tex]\(64x^{14}y^{-3}\)[/tex] can be rewritten as:
[tex]\[
\frac{64x^{14}}{y^3}
\][/tex]
### Step 4:
Combine all the steps to arrive at the final simplified expression:
- The final simplified result is:
[tex]\[
\frac{\left(4x^5y^2\right)^3}{xy^9} = \frac{64x^{14}}{y^3}
\][/tex]
This detailed, step-by-step explanation shows all the justifications required for the given problem.
