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Chapter 5: Problem 5

Expand the powers of 10 or \(\frac{1}{10}\). $$10^{4}$$

Short Answer

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Step by step solution

01

Understand the Base and Exponent

Identify the base and the exponent in the given expression. Here, the base is 10, and the exponent is 4.
02

Interpret the Meaning of the Exponent

The exponent indicates how many times the base (10) is multiplied by itself. In this case, it means multiplying 10 by itself 4 times.
03

Perform the Multiplication

Multiply the base by itself the number of times indicated by the exponent: \[10 \times 10 \times 10 \times 10\]
04

Calculate the Result

Calculate the result of the multiplication: \[10 \times 10 = 100\] \[100 \times 10 = 1000\] \[1000 \times 10 = 10000\]

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Key Concepts

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

Exponents
An exponent is a small number placed to the upper right of a base number. It tells you how many times to multiply the base by itself. For example, in the expression \(10^{4}\), the exponent is 4. This means you multiply 10 by itself 4 times. Exponents are powerful mathematical tools, making it easier to represent large numbers. Instead of writing out long multiplication, you can just use an exponent.
Multiplication
Multiplication is a fundamental math operation where you combine equal groups. When multiplying the same number multiple times, understanding how exponents work is very helpful. For example, calculating \(10 \times 10 \times 10 \times 10\) is possible through basic multiplication, but using exponents simplifies this to \(10^{4}\). Each multiplication step builds on the previous one, leading to more compact expressions through exponents.
Here's how multiplication unfolds with exponents:
  • First Multiplication: \(10 \times 10 = 100\)
  • Second Multiplication: \(100 \times 10 = 1000\)
  • Third Multiplication: \(1000 \times 10 = 10000\)
This step-by-step approach helps to break down the concept.
Base and Exponent
The base is the number that gets multiplied. The exponent tells how many times to multiply the base by itself. In the expression \(10^{4}\), 10 is the base and 4 is the exponent.
Understanding the relationship between base and exponent makes it easier to work with large numbers and simplify mathematical problems. The base remains the same while the exponent indicates the number of times:
  • Base (10) stays the same
  • Exponent (4) shows the multiplication times
This is why \(10^{4}\) results in multiplying 10 by itself 4 times, ultimately simplifying the process and making calculations manageable.

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