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Chapter 3: Problem 89

Evaluate each expression. $$ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} $$

Short Answer

Expert verified
The expression evaluates to \( \frac{8}{27} \).

Step by step solution

01

- Rewrite the Expression

Recognize that \( \frac{2}{3} \) is multiplied by itself three times, which can be written as \( \left( \frac{2}{3} \right)^{3} \).
02

- Apply the Power Rule

Apply the power rule for fractions. Raise both the numerator and the denominator to the power of three:\[ \left( \frac{2}{3} \right)^{3} = \frac{2^{3}}{3^{3}} \].
03

- Calculate the Numerator

Calculate \( 2^{3} = 2 \cdot 2 \cdot 2 = 8 \).
04

- Calculate the Denominator

Calculate \( 3^{3} = 3 \cdot 3 \cdot 3 = 27 \).
05

- Write the Final Fraction

Combine the results from steps 3 and 4 to form the final fraction: \[ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{8}{27} \].

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Key Concepts

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

Exponents
When you see an expression like \(\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}\), you're dealing with exponents. An exponent tells you how many times to multiply a number by itself. In our case, \( \frac{2}{3} \) is multiplied by itself three times. This is written as \( \biggl( \frac{2}{3} \biggl)^{3} \). The small number 3 is the exponent, and it shows how many times the fraction is used in the multiplication.

Here's a simpler example: \( 5^{2}\) means 5 multiplied by itself: \( 5 \times 5 = 25\). The same rule applies to fractions.
The Power Rule
The power rule is a key feature when working with exponents, especially with fractions. According to this rule, you raise both the numerator and the denominator to the power of the exponent. For the fraction \( \frac{2}{3} \), which has an exponent of 3, the power rule allows you to express it as:

\[ \biggl( \frac{2}{3} \biggr)^{3} = \frac{2^{3}}{3^{3}} \]

Therefore, you need to raise both the 2 (numerator) and the 3 (denominator) to the third power separately.
Numerator and Denominator
To complete the problem, calculate the numerator and the denominator separately.

**Numerator**: The top part of the fraction is called the numerator. For our problem, it's 2. Raising 2 to the third power means \(2 \times 2 \times 2 = 8\).
**Denominator**: The bottom part of the fraction is called the denominator. Here, it's 3. Raising 3 to the third power means \(3 \times 3 \times 3 = 27\).

So, \( \biggl( \frac{2}{3} \biggr)^{3} \) becomes \frac{8}{27}\.

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Most popular questions from this chapter

The cost y of producing \(x\) items is, in some cases, expressed as \(y=m x+b .\) The number \(b\) gives the fixed cost tane cost that is the same no matter how many items are produced), and the number \(m\) is the variable cost (the cost of producing an additional item). It costs 2000 dollar to purchase a copier, and each copy costs 0.02 dollar to make. (a) What is the fixed cost? (b) What is the variable cost? (c) Write the cost equation. (d) What will be the cost of producing \(10,000\) copies, based on the cost equation? (e) How many copies will be produced if the total cost is 2600 dollar?

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Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. $$ (6,-3), m=-\frac{4}{5} $$
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