Problem 47 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(a^{-4}\right)^{2}\)

Step-by-Step Solution

Answer

\( \frac{1}{a^8} \)

Step 1: Apply Power Rule

The power rule of exponents states that when raising a power to another power, you multiply the exponents. Given the expression \( \left(a^{-4}\right)^{2} \), apply this rule: \[ a^{-4 \times 2} = a^{-8} \].

Step 2: Rewriting with Positive Exponents

Convert the expression from a negative exponent to a positive exponent by taking the reciprocal. This involves moving the base from the numerator to the denominator: \[ a^{-8} = \frac{1}{a^{8}} \].

Key Concepts

Power Rule of Exponents

The power rule of exponents is a fundamental concept to grasp when dealing with expressions that involve powers. This rule states that when you have a power raised to another power, you simply multiply the exponents. For example, in a problem like \( (a^m)^n \), you can simplify it by rewriting it as \( a^{m \times n} \). Let's look at the exercise you encountered, \( (a^{-4})^2 \). Applying the power rule here means you multiply -4 by 2, resulting in \( a^{-8} \). This step simplifies the expression by reducing the complexity of nested exponents. Remember:

• Multiply the exponents when raising one power to another.
• Keep the base the same as you simplify the expression.

This principle is handy in reducing larger and more complex exponential expressions to simpler forms.

Understanding Negative Exponents

Negative exponents can initially seem confusing, but they have a straightforward concept behind them. A negative exponent indicates a reciprocal. So, when you see \( a^{-n} \), it means \( \frac{1}{a^n} \).In the example from the exercise, once you had the expression \( a^{-8} \), you used the property of negative exponents to convert it into \( \frac{1}{a^8} \). Here's a general reminder of how to handle negative exponents:

• Move the base from the numerator to the denominator to make the exponent positive.
• Reciprocal transformation helps in rewriting the expression without negative exponents.

Understanding this concept helps you rewrite expressions neatly and aids in further simplification.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing the expression to its simplest form while ensuring no negative or zero exponents remain in the final expression.From your exercise, after applying the power rule and addressing negative exponents, you arrived at the expression \( \frac{1}{a^8} \). This expression is now both simplified and uses only positive exponents. When simplifying:

• Consistently apply rules like the power rule to reduce complexity.
• Ensure no negative exponents are left by applying the reciprocal rule.
• Check your work to confirm the expression is as reduced as possible.

By breaking down problems into manageable steps and applying core rules, you can simplify even the most daunting expressions.