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Chapter 7: Problem 25

Simplify by removing a factor equal to 1. $$ \frac{18 t^{3} w^{2}}{27 t^{7} w} $$

Short Answer

Expert verified
The simplified expression is \( \frac{2w}{3t^4} \).

Step by step solution

01

Factorize the numerator and denominator

First, write down the prime factorization of both the numerator and the denominator: Numerator: \[ 18 t^3 w^2 = 2 \times 3^2 \times t^3 \times w^2 \] Denominator: \[ 27 t^7 w = 3^3 \times t^7 \times w \]
02

Cancel out the common factors

Identify and cancel the common factors in the numerator and the denominator: Common factors: \[ 3^2, t^3, w \] Cancel these factors to simplify the fraction: \[ \frac{2 \times 3^2 \times t^3 \times w^2}{3^3 \times t^7 \times w} = \frac{2 \times w}{3 \times t^4} \]
03

Write the simplified form

After canceling the common factors, rewrite the remaining terms to obtain the simplified expression: \[ \frac{2w}{3t^4} \]

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers that have only two distinct positive divisors: 1 and themselves. This method helps in simplifying expressions by revealing the basic building blocks of the numbers involved.
To perform prime factorization:
  • Write down the number.
  • Start dividing by the smallest prime number (2, then 3, 5, etc.) and continue dividing until you can't anymore.
  • Repeat with the result of each division until you end up with 1 or a prime number.
For the exercise:
  • Numerator: The prime factorization of 18 is \(2 \times 3^2\). Combine this with \(t^3\) and \(w^2\).
  • Denominator: The prime factorization of 27 is \(3^3\). Combine this with \(t^7\) and \(w\).
Understanding prime factorization helps in identifying common factors quickly.
Canceling Common Factors
Canceling common factors involves simplifying fractions by dividing the numerator and the denominator by their greatest common divisor. Here's how you can do it:
  • Factorize both the numerator and the denominator into their prime factors.
  • Identify the common factors that appear in both.
  • Divide both the numerator and the denominator by these common factors.
For the exercise:
Identify common factors between \(18 t^3 w^2\) and \(27 t^7 w\). The common factors are \(3^2\), \(t^3\), and \(w\).
Cancel these factors:
  • \( \frac{2 \times 3^2 \times t^3 \times w^2}{3^3 \times t^7 \times w} = \frac{2 \times w}{3 \times t^4} \)
The simplified form no longer contains these common factors, making it easier to work with.
Rational Expressions
Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions is similar to simplifying numerical fractions.
Key steps:
  • Factorize the polynomials in the numerator and the denominator as much as possible.
  • Cancel any common factors between the numerator and the denominator.
  • Rewrite the expression in its simplest form.
From the exercise, we started with the rational expression \( \frac{18 t^3 w^2}{27 t^7 w} \).
By factorizing and canceling common factors, we simplify it to \( \frac{2 w}{3 t^4} \).
Understanding rational expressions is vital for solving algebraic equations, working with functions, and many other areas in higher mathematics.

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