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$Express the following fraction in simplest form, only using positive exponents. ((-4v^(3))^(2))/(12v^(10)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{\left(-4 v^{3}\right)^{2}}{12 v^{10}}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

Q. Express the following fraction in simplest form, only using positive exponents.

\newline

\frac{\left(-4 v^{3}\right)^{2}}{12 v^{10}}

\newline

Answer:

Write Expression: Write down the given expression. $\newline$ We have the expression $\frac{((-4v^{3})^{2})}{12v^{10}}$ .
Apply Power Rule: Apply the power of a power rule to the numerator. $\newline$ The power of a power rule states that $(a^m)^n = a^{m*n}$ . So, we apply this rule to the numerator: $\newline$ $((-4v^{3})^{2}) = (-4)^2 \times (v^{3})^2 = 16 \times v^{3*2} = 16v^{6}$ .
Simplify Denominator: Simplify the denominator. $\newline$ The denominator is $12v^{10}$ . There is no exponent rule to apply here, so it remains as is.
Write Simplified Expression: Write the simplified expression. $\newline$ Now we have the simplified numerator and the original denominator: $\newline$ $(16v^{6})/(12v^{10})$ .
Reduce Fraction: Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of $16$ and $12$ is $4$ . Also, we can subtract the exponents of $v$ since they have the same base when dividing: $(\frac{16}{4})v^{(6-10)} = 4v^{-4}$ .
Express Negative Exponent: Express the negative exponent as a positive exponent. $\newline$ A negative exponent means that the base is on the wrong side of the fraction line, so we move it to the denominator to make the exponent positive: $\newline$ $4v^{-4} = \frac{4}{v^4}$ .

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