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$Express the following fraction in simplest form using only positive exponents. (2d^(4))/(5(d^(4))^(3)) Answer:$

Express the following fraction in simplest form using only positive exponents. $\newline$ $\frac{2 d^{4}}{5\left(d^{4}\right)^{3}}$ $\newline$ 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{2 d^{4}}{5\left(d^{4}\right)^{3}}

\newline

Answer:

Identify Base and Exponents: Identify the base and the exponents in the fraction. $\newline$ We have the fraction $(2d^{4})/(5(d^{4})^{3})$ . The base in the numerator is $d$ with an exponent of $4$ . In the denominator, we have $d$ with an exponent of $4$ raised to the power of $3$ .
Apply Power of Power Rule: Apply the power of a power rule to the denominator. $\newline$ The power of a power rule states that $(a^{m})^{n} = a^{m*n}$ . Therefore, $(d^{4})^{3} = d^{4*3} = d^{12}$ .
Rewrite with Simplified Denominator: Rewrite the fraction with the simplified denominator. $\newline$ Now we have $(\frac{2d^{4}}{5d^{12}})$ .
Simplify by Canceling Factors: Simplify the fraction by canceling out common factors. Since $d^{4}$ is a factor in both the numerator and the denominator, we can divide both by $d^{4}$ . This gives us $\frac{2}{5d^{12-4}} = \frac{2}{5d^{8}}$ .
Check for Further Simplification: Check for any further simplification. There are no common numerical factors between $2$ and $5$ , and the exponents are already positive. Therefore, the fraction is in its simplest form.

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