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Chapter 0: Problem 8

In Exercises \(1-28\), compute the numbers. $$ (.01)^{3} $$

Short Answer

Expert verified
0.000001

Step by step solution

01

Understand the expression

The given expression is \((.01)^{3}\). This means raising \(.01\) to the power of 3. In other words, \(.01\) is multiplied by itself two more times.
02

Express \(.01\) in fractional or decimal form

Recall that \(.01\) can be written as \(\frac{1}{100}\) or just \(10^{-2}\). This helps to simplify the calculation.
03

Calculate the power

Raising \(.01\) to the power of 3 can be written as: \((10^{-2})^3 = 10^{-2 \times 3} = 10^{-6}\).
04

Convert the result back to decimal form

Now \(10^{-6}\) is the same as \(.000001\).

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Key Concepts

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

powers of ten
Power of ten is a term used to express numbers that are the result of raising 10 to an exponent. These powers can be positive or negative, affecting whether the resulting number is large or small. Positive exponents
  • Indicate multiplication by 10. For example, \(10^3\) means 10 multiplied by itself three times: 10 * 10 * 10 = 1000.
  • Create larger numbers.
Negative exponents
  • Serve to divide by 10. For example, \(10^{-2}\) equals \( \frac{1}{10^2} = \frac{1}{100} = 0.01 \).
  • Create smaller numbers.
Using the power of ten helps simplify complex multiplications and divisions, making calculations more manageable and intuitive.
decimal notation
Decimal notation is the standard system for writing numbers and uses a base of 10. The position of the digit determines its value. For example, in the number 0.01, the digit 1 is in the hundredths place, meaning it is \(1 \times 10^{-2}\). When dealing with very small or very large numbers, decimal notation becomes quite handy. Standard format
  • Helps in clear expression and understanding of place value.
  • Keeps calculations straightforward and organized.
Converted forms We can convert decimals into exponential form to simplify calculations. For example, 0.01 can be written as \(10^{-2}\). By using decimals and their exponential forms interchangeably, we can handle mathematical operations more efficiently.
fraction conversion
Fraction conversion involves transforming a decimal into a fraction or vice versa. This helps in better understanding and simplifying mathematical expressions. Conversion to fractions
  • For instance, 0.01 as a fraction is \( \frac{1}{100} \).
  • This involves recognizing that the decimal is a part of a whole and finding its equivalent fraction form.
Conversion to decimals
  • Conversely, fractions can be simplified into decimals for easier calculation. For example, \( \frac{1}{100} \) is equal to 0.01.
  • This is particularly useful in real-world applications where decimals are more commonly used.
Whether dealing with fractions or decimals, understanding how to convert between the two is fundamental in mastering various mathematical concepts.

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