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# Multiplying & dividing powers (integer exponents)

For any base a and any integer exponents n and m, aⁿ⋅aᵐ=aⁿ⁺ᵐ. For any nonzero base, aⁿ/aᵐ=aⁿ⁻ᵐ. These are worked examples for using these properties with integer exponents.

## Want to join the conversation?

• in how did he get 1/4 3
• When you have a negative power, you are taking the reciprocal of the number, and keep the power. So 2^(-2)=1/2^2. So 4^(-3)=1/4^3
• If negative exponents such as 10^-5 is equal to 1/10^5, what would fractions with negative exponents such as 1/10^-5 be equal to?
• Apply the same rule you have cited. As you put it (10)^-5 = 1/(10)^5
The expression in question is 1/(10)^-5.
Lets see! We can write (10)^-5 as 1/(10)^5 (as you wrote).
So 1/(10)^-5 can essentially be written as 1/(1/10^5)
Which is nothing but 10^5 itself( We're basically taking the reciprocal of 1/10^5)

So 1/(10)^-5 =10^5
Cheers!

EDIT: Since perhaps that's a bit long, you can remember it for a general case as:
1/a^-m = a^m
where a and m can be any of positive or negative integers(but not zero!)
Hope that helps!
• How do you divide exponents by exponents? I kinda really don't understand that part.
• An easier way to think about this is to treat the multiplication sign as an addition sign and treat the division sign as a subtraction sign. I'll put an example down below! :)
Therefore, 4_^-3 x 4_^5 is equal to 4_^2.
You would add -3 + 5, which is equal to 2. Then keep the 4 and put the 2 as the exponent!
• how do you do it when both powers are negative and you are multiplying.
• when both powers are negative, and you are multiplying,the negatives cancel eachother out so you would get a positive power.
• For the dividing part, how did you make the exponent of 12^-5 positive and the exponent of x^5 negative?
• The rule for dividing same bases is x^a/x^b=x^(a-b), so with dividing same bases you subtract the exponents. In the case of the 12s, you subtract -7-(-5), so two negatives in a row create a positive answer which is where the +5 comes from. In the x case, the exponent is positive, so applying the rule gives x^(-20-5).
If you want to use two different laws of exponents, you can use the negative exponent rule, if you move an exponent from numerator to denominator (or from denominator to numerator), you have to change the sign. So 12^-5 in the denominator would be the same as 12^5 in the numerator and x^5 in the denominator would be x^-5 in the numerator. Then you would have to use the rule for multiplying same bases shown as x^a * x^b=x^(a+b). Thus, x^-7*x^5 (as moved above) you still get 12^(-7+5) and x^-20 * x^-5 = x^(-20-5).
• what if you don't have the same base? like if you have 5 to the power of 3 times 6 to the power of 2?
• You can't combine the exponents. The bases don't match.
• Isn't multiplying exponents the same as adding exponents then? Because for multiplying exponents you add the exponents and for adding exponents you add the exponents. What is the difference?
• Your terminology is a little off...
If you are multiplying a common base, then you add the exponents. For example: x^7 * x^2 = x^(7+2) = x^11
There is no multiplication of the exponents in this problem. The exponents are beind added. The base values "x" are what is being multiplied.

Multiplying exponents occurs when you have an expression that involves and exponent and that expression is raised to an exponent. For example: (x^7)^2 = x^(7*2) = x^14

Hope this helps.
• What is an integer?
• An integer is a whole number and cannot be a fraction/decimal.
Some examples of numbers that are integers: 3, -403, -7, 1000
Some examples of numbers that are NOT integers: 3.56, -9.41, -30789.99, 0.87