Question

Simplify each expression. $\frac{\left(-5 m n^{2}\right)^{3}}{5 m^{2} n^{4}}$

   Simplify each expression.
$\frac{\left(-5 m n^{2}\right)^{3}}{5 m^{2} n^{4}}$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 8, Problem 40

Instant Answer

Solved by Expert Elizabeth Xu

03/04/2020

Step 1

This gives us $(-5)^{3} \cdot m^{3} \cdot n^{6}$, which simplifies to $-125m^{3}n^{6}$. Show more…

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Simplify each expression. $\frac{\left(-5 m n^{2}\right)^{3}}{5 m^{2} n^{4}}$

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Key Concepts

Exponentiation of a Product

This concept involves raising an entire product to a power. When an expression like a product of constants and variables is raised to an exponent, each factor in the product is raised to that same exponent. This is crucial for rewriting complex expressions in a simpler, spread out form before further simplification is performed.

Laws of Exponents for Division

This principle states that when dividing two expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator. This rule is essential for simplifying rational expressions by reducing the exponents of common bases in the numerator and denominator.

Handling Negative Bases in Exponentiation

When a negative number is raised to an exponent, the sign of the resulting value depends on whether the exponent is even or odd. Specifically, raising a negative base to an odd power yields a negative result, while an even power produces a positive result. This understanding is important to correctly simplify expressions that involve negative factors.

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