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Chapter 0: Problem 5

In Exercises \(1-28\), compute the numbers. $$ (.1)^{4} $$

Short Answer

Expert verified
0.0001

Step by step solution

01

Understand the Problem

The given problem requires finding the value of \((0.1)^{4}\).
02

Convert to Fraction

First, express the decimal as a fraction. Here, 0.1 can be written as \(\frac{1}{10}\).
03

Apply Exponent

Apply the exponent to the fraction: \(\bigg(\frac{1}{10}\bigg)^{4}\). This means raising both the numerator and the denominator to the power of 4.
04

Simplify the Expression

Calculate the result: \(\big(\frac{1}{10}\big)^{4} = \frac{1^{4}}{10^{4}} = \frac{1}{10000}\).
05

Convert Back to Decimal

Convert the fraction back to a decimal: \(\frac{1}{10000} = 0.0001\).

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Key Concepts

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

Decimal to Fraction Conversion
When working with decimals, converting them to fractions can simplify mathematical operations. To convert a decimal to a fraction, identify the place value of the decimal. For example, the decimal \(0.1\) falls in the tenths place. Hence, \(0.1\) converts to \(\frac{1}{10}\). Understanding this step makes other operations easier. Remembering common place values, like tenths, hundredths, and thousandths, helps when converting.
Exponent Rules
Exponentiation involves raising numbers to a power. The crucial rules to remember include:
  • For any non-zero number \(a\), \(a^0 = 1\).
  • \(a^m \times a^n = a^{m+n}\) when multiplying with the same base.
  • \((a^m)^n = a^{m \times n}\) when raising a power to another power.
Using these rules properly ensures accurate results in calculations involving exponents. For example, applying \((\frac{1}{10})^4\) means raising both the numerator (1) and the denominator (10) to the power of 4.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. However, if the numerator is 1, the fraction is already simplest. For example, \(\frac{1}{10000}\) is already simplified as 1 and 10000 have no common divisor other than 1. Simplifying fractions helps in easier interpretation and further calculations.
Decimal Conversion
Converting fractions back to decimals involves division. Take the example \(\frac{1}{10000}\). To convert, we divide 1 by 10000 which yields \(0.0001\). Remembering this method simplifies calculations and helps in interpreting results accurately. Conversions between these forms are fundamental for understanding and solving numerous math problems effectively.

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Most popular questions from this chapter

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\sqrt{x}\left(\frac{1}{4 x}\right)^{5 / 2}\)

Use the laws of exponents to compute the numbers. \(\frac{7^{4 / 3}}{7^{1 / 3}}\)

Use the laws of exponents to compute the numbers. \(\left(6^{1 / 2}\right)^{0}\)

The expressions in Exercises 83-88 may be factored as shown. Find the missing factors. \(2 x^{2 / 3}-x^{-1 / 3}=x^{-1 / 3}(\quad)\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \((9 x)^{-1 / 2}\)
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