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Chapter 4: Problem 94

\(\frac{(8 x)^{-8}(8 x)^{9}}{(8 x)^{13}(8 x)^{-11}}\)

Short Answer

Expert verified
\(\frac{1}{8x}\)

Step by step solution

01

- Simplify the Numerator and Denominator

First, simplify the numerator \( (8x)^{-8}(8x)^9 \) and the denominator \( (8x)^{13}(8x)^{-11} \). Use the properties of exponents \( a^m \times a^n = a^{m+n} \).For the numerator: \[ (8x)^{-8} \times (8x)^9 = (8x)^{-8+9} = (8x)^1 = 8x \]For the denominator: \[ (8x)^{13} \times (8x)^{-11} = (8x)^{13-11} = (8x)^2 \]
02

- Divide the Simplified Terms

Now, divide the simplified numerator by the simplified denominator \( \frac{8x}{(8x)^2} \). Use the property \( a^m / a^n = a^{m-n} \):Simplify: \[ \frac{8x}{(8x)^2} = (8x)^{1-2} = (8x)^{-1} \]
03

- Express with Positive Exponents

Rewrite the expression using positive exponents: \[ (8x)^{-1} = \frac{1}{8x} \]

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Key Concepts

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

simplifying exponents
Simplifying exponents involves combining like terms by following specific properties. When you have a term like \(a^m \times a^n\), you can combine the exponents by adding them together \(a^{m+n}\). This property helps to simplify expressions that might look complex at first.
For example, in our exercise, we simplified the numerator \( (8x)^{-8}(8x)^9 \) by adding the exponents of \(8x \): \( -8 + 9 \). The result was \( (8x)^1 = 8x \).
Similarly, the denominator \( (8x)^{13}(8x)^{-11} \) was simplified by adding the exponents \(13 + (-11) = 2\), giving us \( (8x)^2 \).
multiplication of exponents
Multiplication of exponents is a specific property that helps in combining terms with the same base during multiplication. The rule applied is \(a^m \times a^n = a^{m+n}\). This makes it easier to handle and simplify the expression, regardless of the complexity.
In the step-by-step solution of our exercise, we used this property twice:
  • In the numerator: \( (8x)^{-8} \times (8x)^9 = (8x)^{-8+9} = (8x)^1 = 8x \).
  • In the denominator: \( (8x)^{13} \times (8x)^{-11} = (8x)^{13-11} = (8x)^2 \).

This step helps in converting complex exponent expressions into simpler terms that are easier to manage.
division of exponents
Division of exponents is another important property which guides us in simplifying fractions that have the same base. The rule is \(a^m \/ a^n = a^{m-n}\), meaning you subtract the exponent in the denominator from the exponent in the numerator.
Using this property helps us directly deal with the fraction once we have simplified both parts independently. In the final step of our exercise, we applied this property to get:
  • \( \frac{8x}{(8x)^2} = (8x)^{1-2} = (8x)^{-1}\).

Finally, to express with positive exponents, this becomes \( (8x)^{-1} = \frac{1}{8x} \).

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