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“a to the fifth times a cubed equals a to the eighth” |
Imagine that you have a string of five a’s (all multiplied together, not added), and want to multiply this by a string of three a’s (again, all multiplied together). How many a’s would you end up with?
Write it out:
(a × a × a × a × a) × (a × a × a) = a × a × a × a × a × a × a × a
If you wrote each element of this equation exponentially, it would read:
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“a to the fifth times a cubed equals a to the eighth” |
This leads to the first exponent rule:
When multiplying exponential terms that share a common base, add the exponents.
Other examples:
| Exponentially | Written Out |
| 73 × 72 = 75 | (7 × 7 × 7) × (7 × 7) = 7 × 7 × 7 × 7 × 7 |
| 5 × 52 × 53 = 56 | 5 × (5 × 5) × (5 × 5 × 5) = 5 × 5 × 5 × 5 × 5 × 5 |
| f 3 × f1 = f4 | (f × f × f) × f = f × f × f × f |
Now imagine that you are dividing a string of five a’s by a string of three a’s. (Again, these are strings of multiplied a’s.) What would be the result? Write it out again:
You can cancel out from top and bottom →
→ a × a
If you wrote this out exponentially, it would read:
| a5 ÷ a3 = a2 | “a to the fifth divided by a cubed equals a squared” |
This leads to the second exponent rule:
When dividing exponential terms with a common base, subtract the exponents.
Other examples:
| Exponentially | Written Out |
| 75 ÷ 72 = 73 | (7 × 7 × 7 × 7 × 7) / (7 × 7) = 7 × 7 × 7 |
| 55 ÷ 54 = 5 | (5 × 5 × 5 × 5 × 5) / (5 × 5 × 5 × 5) = 5 |
| f 4 ÷ f1 = f3 | f × f × f × f) / (f) = f × f × f |
These are the first two exponent rules:
| Rule Book: Multiplying and Dividing Like Base with Different Exponents | |
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When multiplying exponential terms that share a common base, add the exponents. a3 × a2 = a5 |
When dividing exponential terms with a common base, subtract the exponents. a5 ÷ a2 = a3 |
Simplify the following expressions by combining like terms.
b5 × b7
(x3)(x4)
These are the most commonly used rules, but there are some other important things to know about exponents.
When something with an exponent is raised to another power, multiply the two exponents together:
If you have four pairs of a’s, you will have a total of eight a’s:
(a × a) × (a × a) × (a × a) × (a × a) = a × a × a × a × a × a × a × a = a8
It is important to remember that the exponent rules just discussed apply to negative exponents as well as to positive exponents. For instance, there are two ways to combine the expression 25 × 2−3:
The first way is to rewrite the negative exponent as a positive exponent, and then combine:
Add the exponents directly:
Simplify the following expressions.
