How to Multiply Power-terms (Exponents) in Algebra

What Are Power-Terms?

Power-terms, also known as exponents, are a way to express repeated multiplication of the same number. They consist of a base and an exponent. The base is the number being multiplied, and the exponent tells us how many times to multiply the base by itself.

Basic Components of Power-Terms

1. Base: The number that is being multiplied.
2. Exponent: The small number written above and to the right of the base, indicating how many times to multiply the base.

For example, in the power-term 2³:

• The base is 2.
• The exponent is 3.

This means you multiply 2 by itself 3 times: 2 × 2 × 2 = 8.

Multiplying Power-Terms with the Same Base

When we multiply power-terms with the same base, we add their exponents. Let’s break it down step-by-step.

Basic Rule

If you have two power-terms with the same base, you can add the exponents together. The rule looks like this:

a^m × aⁿ  = a^m+n

Here:

• a is the base.
• m and n are the exponents.

Example 1

Let’s look at an example to make this clearer.

2³ × 2⁴ 

• Here, the base is 2.
• The exponents are 3 and 4.

According to our rule, we add the exponents together:

2³ × 2⁴ = 2^{3 + 4} = 2⁷ 

Why Does This Rule Work?

When we multiply power-terms with the same base, we are essentially multiplying the base by itself repeatedly.

For example, let’s expand 2³ × 2⁴:

2³ = 2 × 2 × 2

2⁴ = 2 × 2 × 2 × 2

When we multiply these together, we get:

(2 × 2 × 2) × (2 × 2 × 2 × 2)

If we count all the 2s being multiplied, we have seven 2s:

2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷

So, 2³ × 2⁴ = 2⁷.

So, 2³ × 2⁴ = 2⁷

Example 2

Here’s another example:

5² × 5³

• The base is 5.
• The exponents are 2 and 3.

Add the exponents:

5² × 5³  = 5²⁺³ = 5⁵

So, 5² × 5³ = 5⁵

Multiplying Power-Terms with Different Bases

If the bases are different, you cannot simply add the exponents. You need to handle each base separately.

Example

3² × 4²

• Here, the bases are 3 and 4.
• The exponents are both 2, but since the bases are different, we can’t combine them.

So we calculate each term separately:

3² = 3 × 3 = 9

4² = 4 × 4 = 16

Then, we multiply the results:

9 × 16 = 144

So, 3² × 4² = 144

Practice Problems

Let’s practice multiplying power-terms.

1. Simplify 7² × 7³.
2. Simplify 10⁴ × 10².
3. Simplify 2⁵ × 2³.
4. Simplify 6¹ × 6⁴.
5. Simplify 8² × 8³ × 8¹.

Answers to Practice Problems

1.  7² × 7³ = 7⁵
2. 10⁴ × 10² = 10⁶
3. 2⁵ × 2³ = 2⁸
4. 6¹ × 6⁴ = 6⁵
5. 8² × 8³ × 8¹ = 8⁶

Conclusion

Now you know how to multiply power-terms in algebra! Remember, when the bases are the same, you just add the exponents. Keep practicing to become more confident in working with exponents.