Problem 52 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Understand the problem

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

Step 2 - 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: \(\left(b^{-4}\right)^2 = b^{-4 \cdot 2}\).

Step 3 - Multiply the Exponents

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

Key Concepts

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’s 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.