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$Express the following fraction in simplest form using only positive exponents. (5c^(6))/((4c^(3))^(4)) Answer:$

Express the following fraction in simplest form using only positive exponents. $\newline$ $\frac{5 c^{6}}{\left(4 c^{3}\right)^{4}}$ $\newline$ Answer:

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

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

\newline

\frac{5 c^{6}}{\left(4 c^{3}\right)^{4}}

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Answer:

Write Expression and Identify: Write down the given expression and identify the base and exponents. $\newline$ The given expression is $(5c^{6})/((4c^{3})^{4})$ . $\newline$ Here, we have a fraction where the numerator is $5c^{6}$ and the denominator is $(4c^{3})$ raised to the power of $4$ .
Apply Power of Power Rule: Apply the power of a power rule to the denominator. $\newline$ According to the power of a power rule, $(a^m)^n = a^{m*n}$ . $\newline$ So, $(4c^{3})^4 = 4^4 \times c^{3*4} = 4^4 \times c^{12}$ .
Calculate Denominator Value: Calculate the value of $4^4$ . $\newline$ $4^4 = 4 \times 4 \times 4 \times 4 = 256$ . $\newline$ So, the denominator becomes $256c^{12}$ .
Rewrite Expression with Denominator: Rewrite the expression with the calculated denominator. $\newline$ The expression now is $(\frac{5c^{6}}{256c^{12}})$ .
Simplify Fraction: Simplify the fraction by dividing the coefficients and subtracting the exponents. $\newline$ Divide the coefficients: $\frac{5}{256}$ remains unchanged as $5$ and $256$ have no common factors. $\newline$ Subtract the exponents: $c^{(6-12)} = c^{-6}$ . $\newline$ The expression now is $\frac{5c^{-6}}{256}$ .
Express Negative Exponent: Express the negative exponent as a positive exponent. $\newline$ According to the negative exponent rule, $a^{-n} = \frac{1}{a^n}$ . $\newline$ So, $c^{-6} = \frac{1}{c^6}$ . $\newline$ The expression now is $\frac{5}{256c^6}$ .
Check Full Simplification: Check if the expression is fully simplified. $\newline$ The expression $\frac{5}{256c^6}$ is fully simplified as there are no common factors between the numerator and the denominator, and all exponents are positive.