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Chapter 8: Problem 87

Simplify each expression. $$\frac{a^{-2} a^{3}}{a^{4}}$$

Short Answer

Expert verified
The simplified expression is \(\frac{1}{a^3}\).

Step by step solution

01

- Combine the Exponents in the Numerator

To simplify the expression \(\frac{a^{-2} a^{3}}{a^{4}}\), start by combining the exponents in the numerator. Use the property of exponents that says \(a^m \times a^n = a^{m+n}\). Here \(a^{-2} \times a^3 = a^{-2+3} = a^{1}\). So, the expression becomes \(\frac{a^1}{a^{4}}\).
02

- Subtract the Exponents

Next, simplify \(\frac{a^1}{a^{4}}\). Use the property of exponents that says \(a^m / a^n = a^{m-n}\). Here \(a^1 / a^4 = a^{1-4} = a^{-3}\).
03

- Express with Positive Exponent

Finally, express \(a^{-3}\) with a positive exponent by using the property that says \(a^{-m} = \frac{1}{a^m}\). So, \(a^{-3} = \frac{1}{a^3}\).

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Key Concepts

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

Exponent Rules
Exponents are shortcuts for repeated multiplication. For example, instead of writing \(a \times a \times a\), you can write \(a^3\). Here are a couple of key rules to remember:
- Product Rule: When multiplying two exponents with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
- Quotient Rule: When dividing two exponents with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
- Power Rule: When raising an exponent to another exponent, multiply the exponents: \((a^m)^n = a^{m \times n}\).
These rules will help you simplify complex expressions with exponents, like in our example.
Negative Exponents
Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. For instance, \(a^{-n} = \frac{1}{a^n}\). This is useful when simplifying expressions that result in negative exponents.

For example, in step 3 of our exercise, we ended up with \(a^{-3}\). Applying the negative exponent rule, we convert it to the more standard form: \(\frac{1}{a^3}\).
This property helps in transforming the results into a positive exponent form, which is often preferred in final answers.
Properties of Exponents
Understanding the properties of exponents can simplify and solve problems more effectively. Here are some essential properties:
- Zero Exponent: Any number raised to the power of zero is 1: \(a^0 = 1\).
- Negative Exponent: As we've seen, a negative exponent indicates a reciprocal: \(a^{-n} = \frac{1}{a^n}\).
- Identity Exponent: Any number raised to the power of 1 is itself: \(a^1 = a\).

In our example, the key properties used were the negative and quotient rules. Knowing these properties can greatly simplify working through such expressions and make solving exponent problems much easier.

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Most popular questions from this chapter

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