Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. R - Algebra Review

Problem R.3.25

Chapter 1, Problem R.3.25

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. (-6x²)³

Verified step by step guidance

Recognize that the expression is a power of a product: \((-6x^{2})^{3}\). According to the exponent rules, \((ab)^{n} = a^{n} b^{n}\), so you can apply the exponent to both \(-6\) and \(x^{2}\) separately.

Apply the exponent to the constant term: \((-6)^{3}\). This means multiplying \(-6\) by itself three times.

Apply the exponent to the variable term: \((x^{2})^{3}\). Use the power of a power rule, which states \((x^{m})^{n} = x^{m \times n}\).

Combine the results from the previous two steps to write the expression as a product of the simplified constant and the simplified variable term.

Write the final simplified expression by multiplying the constants and expressing the variable with the new exponent.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponentiation of a Product

When raising a product to a power, each factor inside the parentheses is raised to that power separately. For example, (ab)^n = a^n * b^n. This rule allows us to simplify expressions like (-6x²)³ by applying the exponent to both -6 and x².

Power of a Power Rule

The power of a power rule states that (a^m)^n = a^(m*n). This means when an exponent is raised to another exponent, you multiply the exponents. In the expression (-6x²)³, the x² raised to the 3rd power becomes x^(2*3) = x^6.

Handling Negative Bases with Exponents

When raising a negative number to an exponent, the sign depends on whether the exponent is even or odd. An odd exponent preserves the negative sign, while an even exponent results in a positive value. For (-6)³, since 3 is odd, the result is negative.

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