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$Express the following fraction in simplest form, only using positive exponents. ((-2r^(-3))^(3))/(10r^(-2)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{\left(-2 r^{-3}\right)^{3}}{10 r^{-2}}$ $\newline$ 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.

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\frac{\left(-2 r^{-3}\right)^{3}}{10 r^{-2}}

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Answer:

Rewrite with positive exponents: Rewrite the expression with positive exponents. $\newline$ We have the expression $((-2r^{-3})^3)/(10r^{-2})$ . To simplify, we need to convert the negative exponents to positive exponents. $\newline$ $((-2r^{-3})^3)/(10r^{-2})$ can be rewritten as $((-2)^3/(r^3))/(10/r^2)$ .
Simplify numerator: Simplify the numerator. $\newline$ We will calculate $(-2)^3$ and simplify the numerator. $\newline$ $(-2)^3 = -2 \times -2 \times -2 = -8$ . $\newline$ So, the expression becomes $\frac{(-8/r^3)}{(10/r^2)}$ .
Multiply by reciprocal: Multiply by the reciprocal of the denominator. $\newline$ To divide by a fraction, we multiply by its reciprocal. So, we will multiply $(-8/r^3)$ by the reciprocal of $(10/r^2)$ . $\newline$ $(-8/r^3) \times (r^2/10) = (-8 \times r^2) / (r^3 \times 10)$ .
Simplify expression: Simplify the expression. $\newline$ Now we simplify the expression by canceling out the common $r$ terms and dividing the constants. $\newline$ $(-8 \times r^2) / (r^3 \times 10) = (-8/10) \times (r^2/r^3)$ .
Reduce fraction and apply rule: Reduce the fraction and apply the exponent rule. $\newline$ We can reduce the fraction $-\frac{8}{10}$ and apply the rule $a^{m}/a^{n} = a^{m-n}$ for the $r$ terms. $\newline$ $(-\frac{8}{10}) \cdot (r^{2}/r^{3}) = (-\frac{4}{5}) \cdot r^{2-3} = (-\frac{4}{5}) \cdot r^{-1}$ .
Rewrite with positive exponents: Rewrite the expression with positive exponents. $\newline$ We need to express the final answer with positive exponents, so we rewrite $r^{-1}$ as $1/r$ . $\newline$ The final expression is $(-\frac{4}{5}) \cdot (\frac{1}{r})$ .