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Answer :
Let's simplify [tex]\(\frac{(6x^5z)^3}{4x^4z^2}\)[/tex] step by step.
### Step 1: Simplify the Numerator [tex]\((6x^5z)^3\)[/tex]
1. Expand the exponent:
- Raise each part inside the parentheses to the power of 3.
- [tex]\((6)^3 = 6 \times 6 \times 6 = 216\)[/tex]
- [tex]\((x^5)^3 = x^{5 \times 3} = x^{15}\)[/tex]
- [tex]\((z)^3 = z^{3}\)[/tex]
So, the numerator becomes:
[tex]\[ 216x^{15}z^3 \][/tex]
### Step 2: Simplify the Fraction
The fraction is now:
[tex]\[\frac{216x^{15}z^3}{4x^4z^2}\][/tex]
1. Simplify the coefficients:
- Divide [tex]\(216\)[/tex] by [tex]\(4\)[/tex]: [tex]\(216 \div 4 = 54\)[/tex]
2. Simplify the powers of [tex]\(x\)[/tex]:
- Use the property of exponents, [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex].
- [tex]\(x^{15} \div x^4 = x^{15-4} = x^{11}\)[/tex]
3. Simplify the powers of [tex]\(z\)[/tex]:
- Use the same property of exponents, [tex]\(\frac{z^a}{z^b} = z^{a-b}\)[/tex].
- [tex]\(z^{3} \div z^2 = z^{3-2} = z^{1}\)[/tex] or simply [tex]\(z\)[/tex]
### Final Simplified Expression
Combine everything:
[tex]\[54x^{11}z\][/tex]
So, [tex]\(\frac{(6x^5z)^3}{4x^4z^2}\)[/tex] simplifies to [tex]\(54x^{11}z\)[/tex].
### Step 1: Simplify the Numerator [tex]\((6x^5z)^3\)[/tex]
1. Expand the exponent:
- Raise each part inside the parentheses to the power of 3.
- [tex]\((6)^3 = 6 \times 6 \times 6 = 216\)[/tex]
- [tex]\((x^5)^3 = x^{5 \times 3} = x^{15}\)[/tex]
- [tex]\((z)^3 = z^{3}\)[/tex]
So, the numerator becomes:
[tex]\[ 216x^{15}z^3 \][/tex]
### Step 2: Simplify the Fraction
The fraction is now:
[tex]\[\frac{216x^{15}z^3}{4x^4z^2}\][/tex]
1. Simplify the coefficients:
- Divide [tex]\(216\)[/tex] by [tex]\(4\)[/tex]: [tex]\(216 \div 4 = 54\)[/tex]
2. Simplify the powers of [tex]\(x\)[/tex]:
- Use the property of exponents, [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex].
- [tex]\(x^{15} \div x^4 = x^{15-4} = x^{11}\)[/tex]
3. Simplify the powers of [tex]\(z\)[/tex]:
- Use the same property of exponents, [tex]\(\frac{z^a}{z^b} = z^{a-b}\)[/tex].
- [tex]\(z^{3} \div z^2 = z^{3-2} = z^{1}\)[/tex] or simply [tex]\(z\)[/tex]
### Final Simplified Expression
Combine everything:
[tex]\[54x^{11}z\][/tex]
So, [tex]\(\frac{(6x^5z)^3}{4x^4z^2}\)[/tex] simplifies to [tex]\(54x^{11}z\)[/tex].
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