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Chapter 4: Problem 39

Simplify the following. $$\left(\frac{3}{2}\right)^{3}+\left(\frac{1}{2}\right)^{3}$$

Short Answer

Expert verified
The simplified expression is \( \frac{7}{2} \).

Step by step solution

01

Calculate Individual Powers

First, we need to calculate \( \left(\frac{3}{2}\right)^{3} \) and \( \left(\frac{1}{2}\right)^{3} \). \[ \left(\frac{3}{2}\right)^{3} = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} = \frac{27}{8} \] \[ \left(\frac{1}{2}\right)^{3} = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]
02

Sum the Results

Now, add the results obtained from Step 1: \[ \frac{27}{8} + \frac{1}{8} = \frac{27 + 1}{8} = \frac{28}{8} \]
03

Simplify the Fraction

Simplify the fraction obtained in Step 2: \[ \frac{28}{8} = \frac{28 \div 4}{8 \div 4} = \frac{7}{2} \]

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Key Concepts

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

Understanding Exponents
Exponents let us express repeated multiplication efficiently. They show how many times a number, known as the base, is multiplied by itself. For instance, in the expression \((\frac{3}{2})^3\), \(\frac{3}{2}\) is the base and 3 is the exponent. This means we multiply \(\frac{3}{2} \times \frac{3}{2} \times \frac{3}{2}\) which equals \(\frac{27}{8}\). Repeated multiplication might seem tedious, but it helps simplify expressions.
  • Base: The number that is being multiplied.

  • Exponent: The number that tells us how many times to multiply the base by itself.


Understanding exponents is essential for simplifying large calculations into manageable expressions. With practice, it becomes a powerful tool in your math toolkit.
Adding Fractions
Adding fractions involves combining two or more fractions into a single fraction. The key to adding fractions is ensuring they have common denominators. The denominators must be the same so you can add the numerators directly.

In our problem, we added the fractions \(\frac{27}{8}\) and \(\frac{1}{8}\). Since they already have the same denominator (8), we simply added the numerators: \[ \frac{27}{8} + \frac{1}{8} = \frac{27+1}{8} = \frac{28}{8} \]
  • Same denominator: Check if the fractions have the same bottom number.

  • Add numerators: Directly add the top numbers.

  • Simplify if possible: See if the resulting fraction can be reduced to its simplest form.


Fractions with different denominators require finding a common denominator first, but when they already match, addition becomes straightforward.
Simplification of Expressions
Simplifying expressions means reducing them to their simplest form, making them easier to work with. For fractions, this involves finding equivalent expressions with smaller numbers. In our example, we obtained the fraction \(\frac{28}{8}\) from addition, which can be further simplified.

To simplify \(\frac{28}{8}\), we found the greatest common factor (GCF) of 28 and 8, which is 4. Divide both the numerator and the denominator by their GCF to get: \[ \frac{28}{8} = \frac{28 \div 4}{8 \div 4}=\frac{7}{2} \]
  • Greatest Common Factor: The largest number that divides both numerator and denominator.

  • Divide: Reduce both parts of the fraction by the GCF.

  • Simplified expression: Resulting fraction is as small as possible.

Simplifying helps in making expressions easier to understand and compare, especially in complex calculations.

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Most popular questions from this chapter

Find the prime factorization of each number. Two students work to prime factor \(120 .\) One student starts by writing 120 as \(12 \times 10 .\) The other student writes 120 as \(24 \times 5\). Finish each prime factorization. Are they the same? Why or why not?

Simplify. See Section 1.7. $$5^{3}$$

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. Two positive numbers, \(a\) and \(b\), are graphed below. Where should the graph of their average lie?

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$3+\frac{1}{2}$$

In your own words, explain why \(\frac{0}{10}=0\) and \(\frac{10}{0}\) is undefined.
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