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Chapter 5: Problem 104

Perform the indicated operations. Express each answer as a fraction reduced to its lowest terms. \(\frac{3^{5}}{3^{6}}+\frac{2^{3}}{2^{6}}\)

Short Answer

Expert verified
\(\frac{11}{24}\)

Step by step solution

01

Use Rules of Exponents

Given that, when the two numbers with the same base are divided, you subtract the exponents. Hence, we subtract the exponents for both individual fractions: \(\frac{3^{5}}{3^{6}} = 3^{(5-6)}\) and \(\frac{2^{3}}{2^{6}} = 2^{(3-6)}\).
02

Simplify Fractions

Now, simplify the individual fractions. This gives \(3^{-1} = \frac{1}{3}\) and \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).
03

Add Simplified Fractions

Add the two fractions obtained after simplification which is \(\frac{1}{3} + \frac{1}{8}\).
04

Compute Common Denominator

To add these fractions, find the least common denominator (LCD), which is 24 in this case. Hence, rewrite each fraction with the LCD: \(\frac{1}{3} = \frac{8}{24}\) and \(\frac{1}{8} = \frac{3}{24}\).
05

Add Fractions with Common Denominator

Now, that the fractions have the same denominator, add the numerators: \(\frac{8}{24} + \frac{3}{24} = \frac{11}{24}\). The final fraction is already in its simplest form, so no further simplification is required.

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