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Chapter 5: Problem 4

Expand the powers of 10 or \(\frac{1}{10}\). $$10^{3}$$

Short Answer

Expert verified
1000

Step by step solution

01

Understand the exponent

The exponent indicates how many times the base (which is 10 in this case) is multiplied by itself. Here, the exponent is 3, which means 10 needs to be multiplied by itself three times.
02

Write out the multiplication

Express the exponentiation as repeated multiplication: \[10^3 = 10 \times 10 \times 10\]
03

Calculate the product

Now, multiply the numbers step-by-step:First, multiply the first two 10s: \[10 \times 10 = 100\]Then, multiply the result by the remaining 10: \[100 \times 10 = 1000\]
04

Confirm the result

The final result of multiplying 10 by itself three times is 1000.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied, while the exponent tells us how many times the base is multiplied by itself. For instance, in the expression \(10^3\), 10 is the base and 3 is the exponent. This means that 10 is multiplied by itself three times: \[10^3 = 10 \times 10 \times 10 = 1000\].
This method gives us a simpler and more concise way to express repeated multiplication. Exponentiation can have large implications in various fields, like in computing and sciences. Understanding this basic concept helps build a strong foundation for more advanced mathematical studies.
Multiplication
Multiplication is a fundamental arithmetic operation where we find the total of one number taken multiple times. When applying multiplication to exponentiation, the base is taken repeatedly as indicated by the exponent. For example, with the base 10 and exponent 3, we have: \[10 \times 10 \times 10 = 1000\].
We break down the operation step-by-step:
  • First, calculate \(10 \times 10 = 100\)
  • Next, take the result and multiply it again by 10: \[100 \times 10 = 1000\]
This systematic multiplication helps simplify the process of understanding and computing larger numbers.
Base and Exponent
The base and exponent are key components of exponentiation. The base is the number being repeatedly multiplied, and the exponent indicates how many times the base is used in the multiplication.
In the expression \(10^3\), 10 is the base and 3 is the exponent. This translates to multiplying 10 by itself three times.
More generally, \(a^n\) means 'a' (the base) is multiplied by itself 'n' times, where 'n' is the exponent. This concept is essential in simplifying expressions and understanding powers of large numbers. As you master base and exponent concepts, computations like handling scientific notation and complex equations become more accessible.

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Most popular questions from this chapter

For Exercises \(85-90,\) refer to the table. The table gives the closing stock prices (in dollars per share) for the first day of trading for the given month. $$\begin{array}{l|c|c|c|c|c} \text { Stock } & \text { January } & \text { February } & \text { March } & \text { April } & \text { May } \\ \hline \text { IBM } & \$ 97.27 & \$ 99.00 & \$ 92.27 & \$ 95.21 & \$ 103.17 \\\ \hline \text { FedEx } & \$ 109.77 & \$ 111.89 & \$ 114.16 & \$ 106.07 & \$ 106.11 \\ \hline \end{array}$$ By how much did the IBM stock increase between January and May?

Perform the indicated operations. $$0 . \overline{3} \cdot 0.3+3.375$$

What is the greatest number with three places to the right of the decimal that can be created from the digits 2, 9, and 7? Assume that the digits cannot be repeated.

Which number properly represents 3.499999 rounded to the thousandths place? a. 3.500 b. 3.5 c. 3.500000 d. 3.499

For Exercises \(85-90,\) refer to the table. The table gives the closing stock prices (in dollars per share) for the first day of trading for the given month. $$\begin{array}{l|c|c|c|c|c} \text { Stock } & \text { January } & \text { February } & \text { March } & \text { April } & \text { May } \\ \hline \text { IBM } & \$ 97.27 & \$ 99.00 & \$ 92.27 & \$ 95.21 & \$ 103.17 \\\ \hline \text { FedEx } & \$ 109.77 & \$ 111.89 & \$ 114.16 & \$ 106.07 & \$ 106.11 \\ \hline \end{array}$$ Between which two consecutive months did the FedEx stock increase the most? What was the amount of increase?
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