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Chapter 1: Problem 26

Simplify. $$\left(-3 a^{2} b^{-5}\right)^{3}$$

Short Answer

Expert verified
\(-27a^6b^{-15}\)

Step by step solution

01

Apply the Power Rule

When raising a power to another power, multiply the exponents. So, for \((-3 a^{2} b^{-5})^{3},\)apply the rule \((x^m)^n = x^{m imes n}\). This means we need to apply the exponent 3 to each element inside the parentheses:\((-3)^3, (a^2)^3, (b^{-5})^3.\)
02

Simplify Each Component

Let's simplify each part individually:1. \((-3)^3 = -27,\)2. \((a^2)^3 = a^{2 imes 3} = a^6,\)3. \((b^{-5})^3 = b^{-5 imes 3} = b^{-15}.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The Power Rule is a fundamental principle in exponentiation, especially useful when dealing with expressions that involve raising a power to another power. When you see an expression like \((x^m)^n\), this rule instructs you to multiply the exponents, resulting in \((x^{m \cdot n})\). It's a reliable shortcut that simplifies the process of dealing with nested exponents.Let's apply this to the expression \((-3 a^{2} b^{-5})^{3}\):
  • For \(a^2\) raised to the power of 3, use the power rule: \(a^{2 \cdot 3} = a^{6}\).
  • Similarly, \(b^{-5}\) raised to the power of 3 becomes \(b^{-5 \cdot 3} = b^{-15}\).
  • The number \(-3\) is also raised to the power of 3, resulting in \((-3)^3 = -27\).
By applying the Power Rule to each component separately, you obtain a simplified yet precise expression.
Simplifying Expressions
Simplifying expressions is a crucial skill in algebra that involves rewriting expressions in their simplest form. This process often requires a combination of mathematical rules and properties, such as the Power Rule.
When dealing with complex expressions, as in \((-3 a^{2} b^{-5})^{3}\), it's helpful to break down each part of the expression to solve it step-by-step.With this expression:
  • Raise \-3\ to the power of 3 to get \(-27\).
  • Use the Power Rule to simplify \(a^2\) into \(a^{6}\).
  • Simplify \(b^{-5}\) to \(b^{-15}\).
Once simplified, the expression becomes \(-27a^6b^{-15}\). Ensuring each element adheres to established mathematical rules allows for a neat and manageable solution.
Negative Exponents
Negative exponents can be intimidating, but understanding their meaning makes working with them straightforward. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. In other words, \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\).In the context of our expression \((-3 a^{2} b^{-5})^{3}\):
Once the exponents are simplified, as seen in \(-27a^6b^{-15}\), the term \(b^{-15}\) indicates that it's on the wrong side of a fraction line. To simplify, rewrite it as \( rac{1}{b^{15}}\) without altering the other terms:
  • Thus, the final simplified expression becomes \(-27a^6 \cdot \frac{1}{b^{15}}\).
  • This demonstrates the concept that negative exponents flip the base to the opposite side of a fraction.
By mastering negative exponents, expressions become less daunting, allowing you to approach them with confidence.

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