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Chapter 10: Problem 245

In the following exercises, simplify. $$\left(\frac{3}{2}\right)^{5}$$

Short Answer

Expert verified
\frac{243}{32}

Step by step solution

01

Understand the Exponentiation

The expression \(\frac{3}{2}\)^{5} means that the fraction \(\frac{3}{2}\) is multiplied by itself 5 times.
02

Multiply the Fractions

First, write out the multiplication: \(\frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2}\).
03

Multiply the Numerators

To simplify, start by multiplying the numerators (top numbers) together: \(3 \times 3 \times 3 \times 3 \times 3 = 3^{5} = 243\).
04

Multiply the Denominators

Next, multiply the denominators (bottom numbers) together: \(2 \times 2 \times 2 \times 2 \times 2 = 2^{5} = 32\).
05

Combine the Results

Combine the results of the numerators and denominators to form the simplified fraction: \(\frac{243}{32}\).

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Key Concepts

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

Simplifying Fractions
When we simplify fractions, the goal is to make the fraction as simple as possible. This involves reducing the fraction to its lowest terms. Here's a quick way to do it:
  • Find the greatest common divisor (GCD) of both the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCD.
For the fraction \(\frac{243}{32}\), since 243 and 32 have no common factors other than 1, the fraction is already simplified.
Understanding Exponent Rules
Exponentiation involves raising a number to a certain power. This means multiplying the number by itself a specific number of times. The basic rules of exponents can simplify complex problems:
  • Product of Powers: \[a^m \times a^n = a^{m+n}\] (adding the exponents when multiplying like bases).
  • Quotient of Powers: \[\frac{a^m}{a^n} = a^{m-n}\] (subtracting the exponents when dividing like bases).
  • Power of a Power: \[(a^m)^n = a^{mn}\] (multiplying the exponents when raising a power to another power).
In the example, \((\frac{3}{2})^5\), we applied the power of a product rule: \[\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\] to multiply the fraction by itself five times.
Multiplying Fractions
Multiplying fractions may seem tricky, but the steps are straightforward:
  • First, multiply the numerators (top numbers) of the fractions to get the new numerator.
  • Second, multiply the denominators (bottom numbers) to get the new denominator.
For instance, when multiplying \((\frac{3}{2})^5\), write it out as \((\frac{3}{2}) \times (\frac{3}{2}) \times (\frac{3}{2}) \times (\frac{3}{2}) \times (\frac{3}{2})\). Then, multiply the numerators: \[3 \times 3 \times 3 \times 3 \times 3 = 243.\] Next, multiply the denominators: \[2 \times 2 \times 2 \times 2 \times 2 = 32.\] Thus, the resulting fraction is \frac{243}{32}\.

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