Given the following problem, explain and justify the mathematical steps that were performed. If you cannot remember the name of the exponent rule being used, you can explain it in words. Be sure to LABEL your steps \#1-4 with the matching number as the work.

Given problem:
[tex]\[ \frac{(4x^5y^2)^3}{xy^9} \][/tex]

Step 1:
[tex]\[
\frac{4^3 x^{15} y^6}{xy^9}
\][/tex]

Step 2:
[tex]\[
\frac{64 x^{15} y^6}{xy^9}
\][/tex]

Step 3:
[tex]\[
64 x^{14} y^{-3}
\][/tex]

Step 4:
[tex]\[
\frac{64 x^{14}}{y^3}
\][/tex]

Explain and justify each step in your answer.



Answer :

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.