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$Express the following fraction in simplest form, only using positive exponents. (12m^(-7)y^(-9))/((4m^(5))^(-1)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{12 m^{-7} y^{-9}}{\left(4 m^{5}\right)^{-1}}$ $\newline$ 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{12 m^{-7} y^{-9}}{\left(4 m^{5}\right)^{-1}}

\newline

Answer:

Write and Identify: Write down the given expression and identify the negative exponents. $\newline$ The given expression is $(12m^{-7}y^{-9})/((4m^{5})^{-1})$ . $\newline$ We need to convert negative exponents to positive exponents.
Apply Negative Exponent Rule: Apply the negative exponent rule, which states that $a^{-n} = \frac{1}{a^n}$ , to the terms with negative exponents. $\newline$ For the numerator, we have $m^{-7}$ which becomes $\frac{1}{m^7}$ and $y^{-9}$ which becomes $\frac{1}{y^9}$ . $\newline$ For the denominator, we have $(4m^{5})^{-1}$ which becomes $\frac{1}{4m^{5}}$ .
Rewrite with Positive Exponents: Rewrite the expression with positive exponents. $\newline$ The expression becomes: $\newline$ $(12 \times \frac{1}{m^7} \times \frac{1}{y^9}) / (\frac{1}{4m^{5}})$
Simplify by Multiplying: Simplify the expression by multiplying the numerator and the denominator by the reciprocal of the denominator. $\newline$ This means we multiply by $(4m^{5})/1$ . $\newline$ The expression becomes: $\newline$ $(12 \times 1/(m^7) \times 1/(y^9)) \times (4m^{5})/1$
Perform Multiplication: Perform the multiplication. $\newline$ Multiply $12$ by $4$ to get $48$ . $\newline$ Multiply $m^5$ by $1/m^7$ to get $m^{(5-7)}$ which is $m^{-2}$ or $1/m^2$ . $\newline$ The $y^9$ in the numerator remains unchanged. $\newline$ The expression becomes: $\newline$ $(48 \times 1/m^2 \times 1/y^9)$
Rewrite with Positive Exponents: Rewrite the expression with positive exponents. $\newline$ The expression becomes: $\newline$ $\frac{48}{m^2y^9}$