There are several operations that you can perform on exponents. We will try to cover all of them here.
Multiplying Exponents with Same Base
Rule: Add the exponents together and leave the base the same. This applies to exponents with the same base only!
Example: 6³ * 6² = 6^5
Why? Because re-written, it is saying that (6)(6)(6) * (6)(6) = (6)(6)(6)(6)(6)!
To Remember: When multiplying exponents of the same base, add the exponents. If the bases are different, do nothing!
Multiplying Exponents with Different Bases
Rule: You cannot combine the base or exponents, except in one case: the exponents are identical.
Example: x³ * y³ = (xy)³
Why: The above example is saying "3 values of X times 3 values of Y," which is the same as saying "3 values of X times Y."
To Remember: If the exponents varied, such as x² * y³, this cannot be simplified!
The same rules apply as with multiplication, but instead of adding the exponent values, you will subtract!
6³ ÷ 6² = 6¹, or 6.
Why: If you have three values, and you divide by 2 of those values, you are left with 1!
To Remember: For division, subtract instead of add.
Adding and Subtracting Exponents
There really are no rules here: the exponents are not affected.
4³ + 4 ² = 4 ³ ÷ 4 ²
Why: (4)(4)(4) + (4)(4). This is simple addition: by manipulating the exponents (which are a form of multiplication,and thus, division), you would create a drastically different value! The answer is (64) + (16) = 70.
To Remember: Do not try combining bases or exponents for addition and subtraction problems. If we did, we would end up with 4³ + 4 ² = 4^5, which is (4)(4)(4)(4)(4), to equal 1024! Very wrong!
That's all there is to it!
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