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Chapter 6: Problem 14

Multiply. Write each answer in lowest terms. \(\frac{z+9}{12} \cdot \frac{3 z^{2}}{z+9}\)

Short Answer

Expert verified
The simplified answer is \(\frac{z^2}{4}\).

Step by step solution

01

Simplify the Expression

First, observe the multiplication of the two fractions \(\frac{z+9}{12} \times \frac{3z^2}{z+9}\). Notice that \(z+9\) is present in both the numerator of the first fraction and the denominator of the second fraction.
02

Cancel Common Terms

Cancel out the \(z+9\) in both fractions. The expression reduces to \(\frac{1}{12} \times 3z^2\).
03

Multiply the Remaining Terms

Multiply the numerators and the denominators: \(\frac{1 \times 3z^2}{12 \times 1} = \frac{3z^2}{12}\).
04

Reduce to Lowest Terms

Simplify \(\frac{3z^2}{12}\) by dividing both the numerator and the denominator by their greatest common divisor (3): \(\frac{3z^2 \div 3}{12 \div 3} = \frac{z^2}{4}\).

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Key Concepts

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

Fraction Multiplication
When multiplying fractions, you don鈥檛 need to find a common denominator. Instead, just multiply the numerators together and the denominators together. For instance: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \) Remember to simplify the fractions if possible.
Reducing Fractions
Reducing, or simplifying, fractions means making the fraction as simple as possible. It involves dividing the numerator and the denominator by their greatest common divisor (GCD). For example, \( \frac{6}{9} \) can be simplified to \( \frac{2}{3} \), as both 6 and 9 are divisible by 3. Always look to simplify your results for clear and concise answers.
Common Factors in Algebra
Common factors in algebra allow you to simplify expressions. A common factor is a term that divides evenly into each term in an expression. For instance, in the expression \( \frac{z+9}{12} \times \frac{3z^2}{z+9} \), notice the \( z+9 \) in both the numerator and the denominator. This common factor can be canceled out to simplify the multiplication process.

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