Exponent Math

Exponent math

In this unit, we talk about how to multiply, divide, add, and subtract numbers that have exponents. (For a review of the basics of exponents, click here.)

Multiplication: When multiplying two numbers with base 10 with exponents, multiply the coefficients (numbers in front of the 10^exponents) and then add the exponents.

For example (both numbers have coefficients of 1):

• 10⁵ * 10³ = (1 x 10⁵) * (1 x 10³)
• = (1 * 1) * 10^{5 + 3}
• = 1 x 10⁸
• = 10⁸

Example #2 (both numbers have coefficients of 1):

• 10¹³ * 10²²
• = 10^{13 + 22}
• = 10³⁵

Another example (both numbers have non-one coefficients):

• (3 x 10⁸) * (5 x 10⁷)
• = (3 x 5) * (10⁸ x 10⁷)
• = 15 x (10⁸ x 10⁷)
• = 15 x (10^{8 + 7})
• = 15 x 10¹⁵
We can divide 15 by 10 and multiply 10¹⁵ by 10 (multiplying by 10 and dividing by 10 is the same as multiplying by 1: the value will not change). This gives us:


• (15/10) x (10¹⁵ * 10¹)
• = 1.5 x 10^{(15 + 1)}
• = 1.5 x 10¹⁶

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Division: When dividing two numbers with base 10 with exponents, first divide the coefficients then subtract the exponents.

For example:

• 10⁸ / 10⁷ = (1 x 10⁸) / (1 x 10⁷)
• = (1 x 1) * 10^8-7
• = 1 x 10¹ = 10

Another example:

• 10²⁷ / 10¹⁵
• = 10^27-15
• = 10¹²

If there are non-one coefficients, then divide the coefficients also.

For example:

• (3 x 10⁸) / (5 x 10⁶)
• = (3 / 5) * (10⁸ / 10⁶)
• = 0.6 x 10^(8-6)
• = 0.6 x 10²
To get rid of the decimal place, we can multiply 6 by 10, then divide 10² by 10. This is the same as multiplying by 1:


• = (0.6 x 10) * (10²)/10)
• = 6 * 10^{(2 - 1)}
• = 60

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Addition/Subtraction: When you are adding or subtracting two or more numbers with 10 to an exponent, you first need to make sure the exponents are the same in all numbers. Then add/subtract the coefficients and keep the common 10^x.

Addition example: 

• 3 x 10⁸ + 5 x 10⁷ 
We can either make both exponents "7" or "8." We'll convert 3 x 10⁸ to something to 10⁷.

To get 10⁷, we needed to divide 10⁸ by 10. This means that we need to multiply 3 by 10, to get 30. (If we multiply by something, we must divide by the same number. This (*10/10) is the same as multiplying by one.)


• = 30 x 10⁷ + 5 x 10⁷
Now we can just add the coefficients, keeping the same base/exponent:


• = 35 x 10⁷

Subtraction example:

• 6 x 10¹⁰ - 4 x 10⁹ 
First, we need to get the same exponents for each number. We can choose to make both exponents 9 or 10.

We'll make both exponents 9. To get this, we'll first divide 6 x 10¹⁰ by 10 to get 6 x 10⁹. But since we divided by 10, we need to multiple by 10 (so that the number stays the same).
So, this gives us 60 x 10⁹

Now that both exponents are 9, so we can do the subtraction:


• 60 x 10⁹ - 4 x 10⁹ 
Then, we just subtract the coefficients, 60 - 4 (and keep the exponents the same):


• = 56 x 10⁹ 

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Tip: When converting a number to a different exponent, you need to make the coefficient the same amount larger or smaller. For example, if you make the exponent 2 decimal places larger, you need to make the coefficient 2 decimal places smaller. If you make the exponent 5 places smaller, you must make the coefficient 5 places larger.