Question

Simplify. $\frac{\left(-5 m^{7}\right)^{3}}{\left(5 n^{12}\right)^{2}}$

   Simplify.
$\frac{\left(-5 m^{7}\right)^{3}}{\left(5 n^{12}\right)^{2}}$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Sherri Messersmith 2nd Edition

Chapter 2, Problem 26

Instant Answer

Solved by Expert Brianna Orr

07/03/2020

Step 1

This means we multiply the exponents when a power is raised to another power. So, we have: $$\frac{\left(-5 m^{7}\right)^{3}}{\left(5 n^{12}\right)^{2}} = \frac{-5^{3} m^{7 \cdot 3}}{5^{2} n^{12 \cdot 2}}$$ Show more…

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Simplify. $\frac{\left(-5 m^{7}\right)^{3}}{\left(5 n^{12}\right)^{2}}$

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Key Concepts

Simplification of Numerical Coefficients

This involves computing the powers of numerical coefficients separately and then simplifying them, often by dividing common factors. This concept is fundamental in reducing complex rational expressions to their simplest form.

Exponentiation of a Product

This concept states that when a product of terms is raised to an exponent, each term in the product is raised to that power separately. It allows you to distribute the exponent over numerical coefficients and variables, thereby simplifying complex expressions.

Power of a Power Rule

This rule explains that when an already exponentiated base is raised to another power, the exponents multiply. It is essential for reducing nested exponents into a single exponent, which is crucial for simplifying expressions with variables raised to an exponent.

Handling Negative Bases with Exponents

Understanding how negative numbers behave when raised to powers is important. For example, raising a negative number to an odd exponent yields a negative result, while an even exponent results in a positive outcome. This determines the overall sign of the term.

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