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Chapter 0: Problem 30

Evaluate each expression. $$ \left(\frac{7}{10}\right)^{3} $$

Short Answer

Expert verified
\( \left(\frac{7}{10}\right)^{3} = \frac{343}{1000} \)

Step by step solution

01

Understand the Problem

The task is to evaluate the expression \(\left(\frac{7}{10}\right)^{3}\). This means raising the fraction \(\frac{7}{10}\) to the power of 3.
02

Apply the Exponent to Both Numerator and Denominator

Recall that when raising a fraction to a power, both the numerator and the denominator are raised to that power. So,\[\left(\frac{7}{10}\right)^{3} = \frac{7^{3}}{10^{3}}\].
03

Calculate the Powers

Now calculate \(7^{3}\) and \(10^{3}\). \[7^{3} = 7 \times 7 \times 7 = 343\]\[10^{3} = 10 \times 10 \times 10 = 1000\].
04

Write the Final Fraction

Combine the results from the previous step to get the evaluated expression: \[\left(\frac{7}{10}\right)^{3} = \frac{343}{1000}\].

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Key Concepts

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

Fraction Exponents
When dealing with fraction exponents, it's important to understand how the exponent affects both parts of the fraction: the numerator and the denominator. An exponent tells you how many times to multiply a number by itself. For example, raising a fraction like \(\frac{7}{10}\) to the power of 3 means we will multiply \(\frac{7}{10}\) by itself three times. It is crucial to apply the exponent to both the numerator and the denominator separately. This means \(\frac{a}{b}^n\) becomes \(\frac{a^n}{b^n}\). This method helps to manage and simplify complex fraction exponent problems.
Numerator and Denominator Powers
When you raise a fraction to a power, you need to separately raise both the numerator and the denominator to that power. For example, in the problem \(\frac{7}{10}^3\), you must individually calculate \7^3\ and \10^3\.
  • Numerator Powers: Calculate \7^3 = 7 \times 7 \times 7 = 343\.
  • Denominator Powers: Calculate \10^3 = 10 \times 10 \times 10 = 1000\.
Understanding this separation helps you solve and simplify expressions with fraction exponents correctly. Always remember to apply the power to both parts of the fraction.
Power Calculation
To calculate powers, especially for larger numbers, break down the steps to make the process simple. For instance, in the expression \7^3\, calculate it step by step as follows:
  • First, multiply 7 by 7 to get 49.
  • Then, take that result and multiply it by 7 again \49 \times 7 = 343\.
For the denominator, calculating \10^3\ is straightforward as it involves powers of 10:
  • 10 \times\ 10 \times\ 10 = 1000.
Combining these results, the evaluated fraction is \ \(\frac{343}{1000}\). Breaking it down step by step makes power calculations much more manageable.

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