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Chapter 5: Problem 7

Expand the powers of 10 or \(\frac{1}{10}\). $$\left(\frac{1}{10}\right)^{2}$$

Short Answer

Expert verified
\(\frac{1}{100}\)

Step by step solution

01

Identify the expression

The given expression is \ \left(\frac{1}{10}\right)^{2}\. This represents \(\frac{1}{10}\) raised to the power of 2.
02

Apply the power rule

According to the power rule, \(\left(a^m\right)^n = a^{mn}\). Here, \(a\) is \(\frac{1}{10}\) and \(n\) is 2. So, \(\left(\frac{1}{10}\right)^{2} = \frac{1^2}{10^2}\).
03

Simplify the fraction

Now, simplify the expression: \(\frac{1^2}{10^2} = \frac{1}{100}\).

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Key Concepts

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

exponentiation
Exponentiation is a fundamental operation in mathematics where a number, called the base, is raised to a power, or exponent. This indicates how many times the base is multiplied by itself. For instance, in \(\frac{1}{10}^2\), the base is \(\frac{1}{10}\) and the exponent is 2. This means we multiply \(\frac{1}{10}\) by itself. To break this down easier:
  • Identify the base and exponent. Here, the base is \(\frac{1}{10}\) and the exponent is 2.
  • Use the power rule \((a^m)^n = a^{mn}\) which tells us that \((\frac{1}{10})^2 = \frac{1^2}{10^2}\).
  • Simplify to get \(\frac{1}{100}\).
This rule makes it easier to handle complex expressions involving powers, especially fractions.
fractions
Fractions are a way to represent parts of a whole number. A fraction is written as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. When dealing with fractions in exponentiation, like \(\frac{1}{10}^2\), the exponent applies to both the numerator and the denominator. Let's see how:
  • Identify the fraction: \( \frac{1}{10} \). The numerator is 1 and the denominator is 10.
  • Raise both the numerator and denominator to the power indicated: \(\frac{1}{10}^2 = \frac{1^2}{10^2}\).
  • Simplify the results: \(\frac{1^2}{10^2} = \frac{1}{100}\).
Understanding how to work with fractions in this way ensures you can manage more complex problems.
simplifying expressions
Simplifying expressions helps to make them easier to understand and solve. Starting from the expression \((\frac{1}{10})^2\), here's how to simplify step by step:
  • Break down the expression using the power rule: \( (a^m)^n = a^{mn} \).
  • Apply it specifically: \( (\frac{1}{10})^2 = \frac{1^2}{10^2} \).
  • Calculate the results: \( 1^2 = 1 \) and \( 10^2 = 100 \).
  • Combine the simplified parts: \( \frac{1^2}{10^2} = \frac{1}{100} \).
This shows the importance of carefully following each step and applying mathematical rules correctly.
Simplifying expressions not only makes them easier to read but also ensures clarity in solving further math problems.

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Most popular questions from this chapter

Perform the indicated operations. $$(0 . \overline{7}-0 . \overline{6}) \cdot 5.4$$

For Exercises \(59-66,\) simplify the expressions. $$-0.2^{2}$$

Perform the indicated operations. Round the answer to the nearest hundredth when necessary. $$6.5 \div\left(-\frac{3}{5}\right)$$

For Exercises \(59-66,\) simplify the expressions. $$(0.4)^{2}$$

Divide. Round the answer to the indicated place value. Use the rounded quotient to check. $$32.1 \div 0.9 \text { hundredths }$$
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