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# Multiplication with Exponents

by Ron Kurtus (revised 18 January 2022)

When you * multiply exponential expressions*, there are some simple rules to follow. If they have the same base, you simply add the exponents.

Note: Thebaseof the exponential expressionxis^{y}xand theexponentisy.

This is also true for numbers and variables with different bases but with the same exponent. You can apply the rules when other numbers are included.

This rule does not apply when the numbers or variables have different bases and different exponents.

Questions you may have include:

- How do you multiply exponents with the same base?
- What about different bases but with same exponent?
- What about with other numbers?
- When does the rule not apply?

This lesson will answer those questions.

## Multiplying exponents with the same base

When you multiply two variables or numbers that have the *same base*, you simply **add** the exponents.

(x^{a})*(x^{b}) = x^{a+b}

Thus **x ^{3}*x^{4} = x^{3+4} = x^{7}**.

Proof: Since

xand^{3}= x*x*xx, then^{4}= x*x*x*x

(x*x*x)*(x*x*x*x) =x*x*x*x*x*x*x =x^{7}

### Demonstration with numbers

A demonstration of that rule is seen when you multiply **7 ^{3}** times

**7**. The result is:

^{2}

(7*7*7)*(7*7) =

7*7*7*7*7 =7^{5}

Instead of writing out the numbers, you can simply add the exponents:

7^{3}*7^{2}= 7^{3+2}= 7^{5}

Likewise, **2 ^{3}*2^{5}*2^{2} = 2^{3+5+2} = 2^{10}**.

You can see that when you multiply numbers of the same base raised to a power, you **add** their exponents.

## Different bases but same exponent

When you multiply two variables or numbers or with *different bases* but with the *same exponent*, you can simply multiply the bases and use the same exponent. For example:

(x^{a})*(y^{a}) = (xy)^{a}

Also:

(x^{3})*(y^{3}) = xxx*yyy = (xy)^{3}

Likewise, with numbers:

3^{2}*4^{2= }(3*4)^{2}= 12^{2}= 144

## Including other numbers

If you have exponential numbers that are multiplied by other numbers, you can easily do the arithmetic. For example, simplify:

(12*7^{5})*(2*7^{3})

Rearrange the numbers:

(12*2)*(7^{5}*7^{3})

Then add the exponents:

24*7^{8}

The other numbers or variables can also be exponentials. Some examples include:

(3^{3}*5^{2})*(5^{3}*3^{3}) = (3^{3+3})*(5^{2+3}) = 3^{6}*5^{5}

(7*x^{3})*(y^{2}*x^{5}) = 7y^{2}x^{8}

(a^{3}*b^{3})*(b^{6}*a^{5}) = a^{8}b^{9}

## When rule does not apply

When you multiply expressions with *different bases and different exponents*, there is no rule to simplify the process.

For example, suppose you want to multiply **2 ^{3}*5^{2}**.

You can see that **2 ^{3} = 8** and

**5**. Thus

^{2}= 25**8*25 = 200**. But, if you tried

**(2*5)**, you would get

^{3+2}**10**, which is incorrect.

^{5}## Summary

When you multiply two numbers or variables with the same base, you simply add the exponents. When you multiply expressions with the same exponent but different bases, you multiply the bases and use the same exponent.

When you include other numbers or variables in the multiplication, you simply break it up into several multiplications, such as **(x*10 ^{5})*(x*10^{3}) = **

**x**.

^{2}*10^{8}When you multiply expressions with different bases and different exponents, there is no rule to simplify the process.

Always do your best

## Resources and references

### Websites

**Exponents: Basic Rules** - PurpleMath.com

**Exponent Rules** - RapidTables.com

**Laws of Exponents** - MathisFun.com

**Exponents Calculator** - CalculatorSoup.com

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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## Multiplication with Exponents