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

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

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

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

\newline

\frac{\left(-3 w^{-3}\right)^{5}}{-6 w^{-5}}

\newline

Answer:

Move Exponents to Opposite Parts: Rewrite the negative exponents as positive exponents by moving the terms with negative exponents to the opposite part of the fraction. $\newline$ $((-3w^{-3})^5)$ becomes $(-3)^5 \times (w^{-3})^5$ in the numerator. $\newline$ $(-6w^{-5})$ becomes $-6 \times w^{-5}$ in the denominator. $\newline$ We can move $w^{-3}$ to the denominator and $w^{-5}$ to the numerator to make the exponents positive.
Apply Power to Factors: Apply the power to each factor inside the parentheses. $\newline$ $(-3)^5 = -243$ because $(-3)$ multiplied by itself $5$ times is $-243$ . $\newline$ $(w^{(-3)})^5 = w^{(-15)}$ because when you raise a power to a power, you multiply the exponents. $\newline$ So, the numerator becomes $-243 \times w^{(-15)}$ . $\newline$ In the denominator, we have $-6 \times w^{(-5)}$ , which we will address in the next step.
Adjust Exponents: Move $w^{-15}$ from the numerator to the denominator and $w^{-5}$ from the denominator to the numerator to make the exponents positive. $\newline$ The expression becomes $-243 \times w^5 / (-6 \times w^{15})$ .
Simplify Fraction: Simplify the fraction by dividing the numerical coefficients and subtracting the exponents of like bases. $\newline$ $-243 / -6 = 40.5$ (This is a math error because $-243$ divided by $-6$ is actually $40.5$ without the decimal point.) $\newline$ $w^5 / w^{15} = w^{5-15} = w^{-10}$ (This is correct, but we need to move $w^{-10}$ to the denominator to make the exponent positive.)