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Answer :
To simplify the expression [tex]\(\left(\frac{192 x^{12}}{3 x^3}\right)^{\frac{1}{3}}\)[/tex], follow these steps:
1. Simplify the fraction inside the parenthesis:
Start with [tex]\(\frac{192 x^{12}}{3 x^3}\)[/tex].
- Divide the coefficients: [tex]\( \frac{192}{3} = 64 \)[/tex].
- For the variables, apply the law of exponents: [tex]\( x^{12} \div x^3 = x^{12-3} = x^9 \)[/tex].
So, the expression inside the parenthesis simplifies to [tex]\(64 x^9\)[/tex].
2. Take the cube root of the simplified expression:
We need to find [tex]\((64 x^9)^{\frac{1}{3}}\)[/tex].
- Take the cube root of the coefficient: [tex]\((64)^{\frac{1}{3}} = 4\)[/tex], because [tex]\(4^3 = 64\)[/tex].
- Apply the cube root to the variable [tex]\(x^9\)[/tex]: [tex]\((x^9)^{\frac{1}{3}} = x^{9 \cdot \frac{1}{3}} = x^3\)[/tex].
Putting these together, the expression simplifies to [tex]\(4 x^3\)[/tex].
So, the fully simplified expression is [tex]\(4 x^3\)[/tex].
1. Simplify the fraction inside the parenthesis:
Start with [tex]\(\frac{192 x^{12}}{3 x^3}\)[/tex].
- Divide the coefficients: [tex]\( \frac{192}{3} = 64 \)[/tex].
- For the variables, apply the law of exponents: [tex]\( x^{12} \div x^3 = x^{12-3} = x^9 \)[/tex].
So, the expression inside the parenthesis simplifies to [tex]\(64 x^9\)[/tex].
2. Take the cube root of the simplified expression:
We need to find [tex]\((64 x^9)^{\frac{1}{3}}\)[/tex].
- Take the cube root of the coefficient: [tex]\((64)^{\frac{1}{3}} = 4\)[/tex], because [tex]\(4^3 = 64\)[/tex].
- Apply the cube root to the variable [tex]\(x^9\)[/tex]: [tex]\((x^9)^{\frac{1}{3}} = x^{9 \cdot \frac{1}{3}} = x^3\)[/tex].
Putting these together, the expression simplifies to [tex]\(4 x^3\)[/tex].
So, the fully simplified expression is [tex]\(4 x^3\)[/tex].
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