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Chapter 5: Problem 52

\(\left(b^{-4}\right)^{2}\)

Short Answer

Expert verified
The simplified expression is \(b^{-8}\).

Step by step solution

01

- Understand the problem

Given the expression \( \left(b^{-4}\right)^2\), you need to simplify it by applying the laws of exponents.
02

- Apply the power of a power rule

When you raise a power to another power, multiply the exponents. The power of a power rule is \(\left(a^m\right)^n = a^{m\cdot n}\). Here, \(a = b\), \(m = -4\), and \(n = 2\). Thus, applying the rule: \(\left(b^{-4}\right)^2 = b^{-4 \cdot 2}\).
03

- Multiply the Exponents

Multiply \(-4\) by \(2\): \(-4 \cdot 2 = -8\). Therefore, \(b^{-4\cdot2} = b^{-8}\).

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Key Concepts

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

Power of a Power Rule
When working with exponents, understanding the power of a power rule is crucial. This rule simplifies expressions where an exponent is raised to another exponent. In general, the rule is expressed as \((a^m)^n = a^{m \cdot n}\).

  • For example, in the given expression \(\left(b^{-4}\right)^2\), you can apply this rule.
  • This means multiplying the exponents together: \(\left(b^{-4}\right)^2 = b^{-4 \cdot 2} = b^{-8}\).
This simplifies complex exponential expressions and makes them easier to work with.

You can always remember this rule as 'multiply the exponents'. Practice it with different numbers to get comfortable using it.
Exponent Multiplication
Exponent multiplication involves simplifying expressions where exponents are multiplied. This rule is part of the power of a power rule.

When you see an expression like \(a^{m \cdot n}\), you multiply the exponents. For instance:
  • Given an expression like \((2^3)^4\).
  • You multiply the exponents \((3\cdot4) = 12\)
  • The result will be \((2^{12}) = 4096\).
This rule helps to combine exponents for large numbers in a simple and efficient way.

It鈥檚 very useful in algebra and higher mathematics.
Negative Exponents
Negative exponents can be confusing at first, but they follow a clear rule. A negative exponent means the reciprocal of the base raised to the opposite positive exponent. For example, \(a^{-m} = \frac{1}{a^m}\).

Let's take a closer look:
  • Consider the expression \(b^{-8}\).
  • According to the rule, it equals \frac{1}{b^8}\.
Using negative exponents can often make expressions simpler and more manageable.

They are useful in solving equations and simplifying mathematical expressions.

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Most popular questions from this chapter

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A drafter is making enlargements of a rectangular drawing that preserve the relative width and length of the drawing. The length of the drawing is five- fourths of the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.
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