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

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

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Multiplication with rational exponents

Full solution

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

\newline

\frac{5 z^{6}}{(4 z)^{4}}

\newline

Answer:

Apply Power Rule: Simplify the expression by applying the power rule for exponents. $\newline$ The power rule states that a*b)^n = a^n * b^n\. In this case, we can apply the rule in reverse to the denominator \$4z)^4\ to separate the exponent over \$4 and $z$ . $\newline$ $(5z^{6})/((4z)^{4}) = (5z^{6})/(4^4 * z^4)$
Calculate Value: Calculate the value of $4^4$ . $\newline$ $4^4 = 4 \times 4 \times 4 \times 4 = 256$ $\newline$ So, the expression becomes $(\frac{5z^{6}}{256 \times z^4})$ .
Use Quotient Rule: Divide the terms with the same base using the quotient rule for exponents. $\newline$ The quotient rule states that $a^n / a^m = a^{(n-m)}$ when n > m. Here, we have $z^6 / z^4$ , which simplifies to $z^{(6-4)} = z^2$ . $\newline$ So, the expression simplifies to $(5z^2)/256$ .
Check for Simplification: Check if the fraction can be simplified further. $\newline$ Since $5$ and $256$ have no common factors other than $1$ , the fraction is already in its simplest form.

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