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Chapter 4: Problem 65

Rewrite each fraction with the indicated denominators. $$\frac{8}{a b}=\frac{ }{a b^{3}}$$

Short Answer

Expert verified
\[ \frac{8}{a b} = \frac{8 b^{2}}{a b^{3}} \]

Step by step solution

01

Identify the Given Fraction

Start with the given fraction \[ \frac{8}{a b} \]. The goal is to rewrite this fraction so that the new denominator is \[ a b^{3} \].
02

Determine the Multiplication Factor

Identify what factor is missing in the original denominator to match the new denominator. Since the new denominator is \[a b^{3} \] and the original is \[a b \], recognize that \[b^{2} \] is the factor that makes \[b \] into \[b^{3} \] (since \[ b \times b^2 = b^3 \]).
03

Multiply Numerator and Denominator

Multiply both the numerator and the denominator of the original fraction by \[ b^{2} \]. Therefore, \[ \frac{8}{a b} \times \frac{b^{2}}{b^{2}} = \frac{8 b^{2}}{a b^{3}} \].
04

Write the Rewritten Fraction

After performing the multiplication, the rewritten fraction with the new denominator \[a b^{3} \] is \[ \frac{8 b^{2}}{a b^{3}} \].

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Key Concepts

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

fractions
Fractions are a way to represent parts of a whole. They consist of two parts: a numerator and a denominator. The numerator is the top part and shows how many parts of the whole are being considered. The denominator is the bottom part and indicates how many parts the whole is divided into. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator. The fraction means 3 parts out of 4 equal parts.
denominators
Denominators are critical in understanding fractions. They tell us the number of equal parts into which the whole is divided. For example, in \(\frac{1}{2}\), 2 is the denominator indicating the whole is split into 2 equal parts. Changing denominators involves making parts bigger or smaller but keeping the value the same. This is done by multiplying or dividing the numerator and the denominator by the same number.
step by step solution
Breaking down problems into steps makes them easier to solve. Let鈥檚 consider our exercise:
  • Start with the given fraction \(\frac{8}{a b}\).

  • Our goal is to rewrite this fraction with the new denominator \(a b^{3}\).

  • We notice that to convert \(a b\) into \(a b^{3}\), we need to multiply by \(b^{2}\) (since \(b \times b^2 = b^3\)).

  • We then multiply both numerator and denominator by \(b^{2}\) making it \(\frac{8}{a b} \times \frac{b^{2}}{b^{2}} = \frac{8 b^{2}}{a b^{3}}\).

  • The fraction is now \(\frac{8 b^{2}}{a b^{3}}\).
multiplication factor
A multiplication factor is a number used to multiply both the numerator and the denominator of a fraction to change its form without altering its value. For example, to change \(\frac{8}{a b}\) to have a denominator of \(a b^{3}\), we need the multiplication factor \(b^2\). By multiplying both the numerator (8) and the denominator \(a b\) by \(b^2\), we maintain the value of the fraction while changing its form to \(\frac{8 b^{2}}{a b^{3}}\). The key is to multiply both parts by the same number so the overall value remains unchanged.

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