Multiplying Exponents is Easy, but Sometimes It’s Even Easier
We’ve already talked some about multiplying exponents with the same base so you know there is always a trick or two handy when multiplying two terms with exponents on them.
Multiplying exponents with different bases is similar, and as you can guess there’s a trick we can use some of the time to make multiplying exponents with different bases (but in this case, with the same power) just a little easier.
Read on for more, then we’ll cover the more general case of multiplying two terms with different exponents and different powers as well.
Multiplying Exponents with Different Bases by the Same Power
You recall that exponents are really just repeated multiplication, so expanding them out to simple multiplication steps is one way to solve an exponent problem. Here’s an example
23 × 33 = (2×2×2) × (3×3×3) = 8×27 = 216
Here when we expanded the exponents out, we grouped the numbers in parenthesis for each expansion, however we can use the associative property of multiplication to move those parenthesis any way we’d like.
In this example, we have exactly three 2’s and exactly three 3’s, which is no coincidence after all because we are specifically looking at terms raised to the same power. Let’s take our expansion, and do a little bit of associative regrouping…
23 × 33 = (2×2×2) × (3×3×3) = (2×3) × (2×3) × (2×3) = 6×6×6 = 216
All we did here was move some parenthesis around a bit, but the answer works out the same. But if you look carefully, you’ll see an interesting pattern that we can exploit to make our exponent multiplication a little more interesting…
23 × 33 = (2×3) × (2×3) × (2×3) = (2×3)3 = 63 = 216
Do you see what we did? All that happened was we turned all those three 2×3 terms back into an exponential term raised to the third power. Because the power was the same in the original terms, we could group the multiplications of the bases together, and raise that combination to the original power to get the result.
The more general way these combine is expressed like this…
ny × my = (n×m)y
What this says is that for any two numbers, n and m, if they are both raised to the same power, y, then the result will be same as if n and m are multiplied and then that product is raised to the y power.
This can be a easy way to simplify two terms if they are both raised to the same power, but what if we have to multiply exponents with different bases and different powers?
Multiplying Exponents with Different Bases and with Different Powers
Unfortunately, there’s no simple trick for multiplying exponents with different bases and with different powers. You just need to work two terms out individually and multiply their values to get the final product…
24 × 33 = (2×2×2×2) × (3×3×3) = 16×27 = 432
