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Chapter 4: Problem 31

Simplify. $$\frac{16}{9} \cdot\left(\frac{1}{2}\right)^{3}$$

Short Answer

Expert verified
\(\frac{2}{9}\)

Step by step solution

01

Evaluate the exponent

Calculate \left(\frac{1}{2}\right)^{3}$$. An exponent means repeated multiplication of the base. So we have: \(\left(\frac{1}{2}\right)^3 = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8}\).
02

Multiply the fractions

Now multiply \(\frac{16}{9}\) by \(\frac{1}{8}\). Multiplying fractions involves multiplying the numerators together and the denominators together: \(\frac{16}{9} \cdot \frac{1}{8} = \frac{16 \cdot 1}{9 \cdot 8} = \frac{16}{72}\).
03

Simplify the fraction

Finally, simplify the fraction \(\frac{16}{72}\). Find the greatest common divisor (GCD) of 16 and 72, which is 8, and divide both the numerator and the denominator by 8: \(\frac{16 \div 8}{72 \div 8} = \frac{2}{9}\).

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Key Concepts

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

Evaluating Exponents
Evaluating exponents is the process of multiplying a number by itself a certain number of times. When we see an exponent, it tells us to repeat the multiplication of the base number. For example, \(\frac{1}{2}^3\) means \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\). Let's break this down:
  • Firstly, we multiply \(\frac{1}{2}\) by \(\frac{1}{2}\), which gives us \(\frac{1 \times 1}{2 \times 2} = \frac{1}{4}\).
  • Secondly, we multiply \(\frac{1}{4}\) by \(\frac{1}{2}\) again, resulting in \(\frac{1 \times 1}{4 \times 2} = \frac{1}{8}\).

Thus, we have calculated \(\frac{1}{2}^3\) as \(\frac{1}{8}\). This process helps in breaking down the exponentiation to simpler multiplication steps.
Multiplying Fractions
Multiplying fractions might seem tricky at first, but it's quite straightforward once you understand the process. To multiply fractions, you just multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For instance, for \(\frac{16}{9} \times \frac{1}{8}\):
  • Multiply the numerators: \16 \times 1 = 16\
  • Multiply the denominators: \9 \times 8 = 72\

So, \(\frac{16}{9} \times \frac{1}{8} = \frac{16}{72}\). This method ensures that your fractions are combined correctly and keeps all parts in proportion.
Finding Greatest Common Divisor
Simplifying fractions requires finding the greatest common divisor (GCD). The GCD of two numbers is the largest number that can divide both of them without leaving a remainder. For example, in the fraction \(\frac{16}{72}\), we need to find the GCD of 16 and 72 to simplify:
  • List the factors of 16: 1, 2, 4, 8, 16
  • List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The greatest common factor they share is 8.
We then divide both the numerator and the denominator by 8:
  • 16 divided by 8 is 2
  • 72 divided by 8 is 9

Therefore, \(\frac{16}{72}\) simplifies to \(\frac{2}{9}\). This process ensures fractions are in their simplest form, making them easier to work with and understand.

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Most popular questions from this chapter

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