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Chapter 4: Problem 43

Write each decimal using a negative exponent. $$0.01$$

Short Answer

Expert verified
0.01 can be expressed as \( 10^{-2} \).

Step by step solution

01

Understand the Problem

We need to express the decimal 0.01 using a negative exponent. This involves rewriting the decimal as a fraction and then using the properties of exponents.
02

Convert the Decimal to a Fraction

The decimal 0.01 can be written as the fraction \( \frac{1}{100} \) because there are two decimal places.
03

Express the Fraction with a Power of Ten

Next, realize that \( \frac{1}{100} \) is the same as \( \frac{1}{10^2} \) since 100 is \( 10^2 \).
04

Use a Negative Exponent to Simplify

The expression \( \frac{1}{10^2} \) can be rewritten using a negative exponent as \( 10^{-2} \).

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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 converting decimals to fractions, it's all about understanding the place value system. Decimals represent parts of a whole number, and each digit after the decimal point signifies a fractional power of ten.
For example, in the decimal 0.01, the digit '1' is in the hundredths place, which translates the decimal to a fraction of \( \frac{1}{100} \). Notably, the number of zeroes in the denominator corresponds to the number of decimal places.
This systematic approach not only allows us to transform any decimal into a fraction but also lays the foundation for expressing numbers in terms of powers of ten, which is our next concept.
Powers of Ten
Powers of ten are incredibly useful because they offer a simplified way to represent and manipulate large numbers or very small numbers, like decimals. They are expressed as \( 10^n \), where \( n \) is an integer.
  • A positive \( n \) means multiplying by ten; for example, \( 10^2 = 100 \).
  • A negative \( n \) indicates division by ten; thus, \( 10^{-2} \) equals \( \frac{1}{100} \).
The decimal 0.01 translates to \( \frac{1}{100} \), and since 100 is \( 10^2 \), we express \( \frac{1}{100} \) as \( 10^{-2} \). By using powers of ten, we simplify our expressions and make calculations more straightforward.
Properties of Exponents
Exponents are powerful tools in mathematics that condense multiplication and division into a simpler form. Understanding their properties is crucial for operations involving powers.
Key properties include:
  • Product of powers: \( a^m \times a^n = a^{m+n} \).
  • Quotient of powers: \( a^m / a^n = a^{m-n} \).
  • Power of a power: \( (a^m)^n = a^{m \times n} \).
  • Negative exponents: \( a^{-n} = \frac{1}{a^n} \).
In our decimal-to-exponent example, we used the negative exponent property. We expressed \( \frac{1}{100} \) as \( 10^2 \) on the denominator and then transformed it to \( 10^{-2} \) using negative exponents. These properties ensure we can simplify and manipulate expressions efficiently.

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Most popular questions from this chapter

Solve each equation. Check your solution. $$16=5 k-4$$

Express each number in scientific notation. $$0.059$$

Find each product or quotient. Express using exponents. $$10 x^{3} y \cdot\left(-2 x y^{2}\right)$$

Find each product or quotient. Express using exponents. $$6 p^{7} \cdot 9 p^{7}$$

Use the Distributive Property to write each expression as an equivalent expression. $$-3(t+4)$$
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