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((6n^(3)*4n^(3))/(5n^(-2)))^(-1)

$\left(\frac{6 n^{3} \cdot 4 n^{3}}{5 n^{-2}}\right)^{-1}$

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Multiplication with rational exponents

Full solution

Q.

\left(\frac{6 n^{3} \cdot 4 n^{3}}{5 n^{-2}}\right)^{-1}

Write Expression: Write down the given expression and apply the power of a quotient rule. $\newline$ The power of a quotient rule states that $(a/b)^n = a^n / b^n$ . $\newline$ Given expression: $((6n^{3}\cdot4n^{3})/(5n^{-2}))^{-1}$ $\newline$ Apply the power of a quotient rule to the entire expression.
Apply Power Rule: Distribute the exponent of $-1$ to both the numerator and the denominator. $\newline$ $\left(\left(6n^{3}\cdot 4n^{3}\right)^{-1}\right)/\left(5n^{-2}\right)^{-1}$
Distribute Exponent: Simplify the numerator and the denominator separately. $\newline$ Numerator: $(6n^{3}\cdot4n^{3})^{-1} = (24n^{6})^{-1} = 24^{-1} \cdot n^{-6}$ $\newline$ Denominator: $(5n^{-2})^{-1} = 5^{-1} \cdot n^{2}$
Simplify Numerator: Combine the simplified numerator and denominator. $\newline$ $(24^{-1} \cdot n^{-6}) / (5^{-1} \cdot n^{2})$
Simplify Denominator: Simplify the expression by multiplying the reciprocal of the denominator with the numerator. $\newline$ $(24^{-1} \cdot n^{-6}) \cdot (5^{1} \cdot n^{-2})$
Combine Numerator & Denominator: Multiply the coefficients and add the exponents of like bases. $\newline$ $(\frac{1}{24} \times 5) \times n^{(-6 - 2)}$ $\newline$ = $(\frac{5}{24}) \times n^{-8}$
Multiply Reciprocals: Write the final simplified expression. $\newline$ The final simplified expression is $(\frac{5}{24}) \cdot n^{-8}$ , which can also be written as $\frac{5}{24n^{8}}$ .

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