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## Multiplication with Exponents

by Ron Kurtus (revised 13 November 2015)

When you * multiply exponential expressions* that have the same base, you simply add the exponents. This is true for both numbers and variables. When you include other numbers in the multiplication, you simply factor and break the expression into several multiplications.

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

This rule does not apply when the numbers or variables have a different base.

Questions you may have include:

- How do you multiply exponential numbers?
- How do you multiply variables raised to a power?
- How do you include other numbers when multiplying?

This lesson will answer those questions.

## Multiplying exponential numbers

When you multiply two numbers or variables with the same base, you **add** the exponents. This rule does not hold if the numbers are of a different base.

### Same base

A demonstration or verification of that rule is seen when you multiply **7*7*7** times **7*7**. The result is:

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

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

But **7*7*7** = **7 ^{3}** and

**7*7**=

**7**. Thus,

^{2}**7**.

^{3}*7^{2}= 7^{3+2}= 7^{5}Also, **2 ^{3}*2^{5}*2^{2} = 2^{3+5+2} = 2^{10}**.

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

### When rule does not apply

This rule does not apply when multiplying exponents of a different base.

For example, you cannot add exponents in **3 ^{2}*4^{2}**. The numbers must be multiplied out as

**3**.

^{2}*4^{2}= 9*16## Multiplying variables

When you multiply two variables with the same base, you add the exponents. You cannot do that when the bases of the exponential numbers are different.

### Same base

Thus **x ^{3}*x^{4} = x^{3+4} = x^{7}**. This can be proved, since

**x**and

^{3}= x*x*x**x**, then

^{4}= x*x*x*x

(x*x*x)*(x*x*x*x) =

x*x*x*x*x*x*x =x^{7}

Also, when both the base and exponents are variables, **(x ^{a})*(x^{b}) = x^{a+b}**.

### Different base

If the base numbers are different, this rule does not apply. For example **(x ^{6})*(y^{3})** cannot be simplified.

## 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}

## Summary

When you multiply two numbers or variables with the same base, you simply add the exponents. This is true for both numbers and variables.

For example, **2 ^{3}*2^{7}= 2^{10}** and

**c**.

^{3}*c^{4}= c^{7}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}Always do your best

## Resources and references

### Websites

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

**Exponent Rules** - RapidTables.com

**Exponents Calculator** - CalculatorSoup.com

### Books

## Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

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## Where are you now?

## Multiplication with Exponents