Save Time by Combining Exponents with the Same Base
You know your order of operations, and you know when you see a couple of terms with exponents in them that you’re supposed to sort all of that out before you take on things like multiplication.
However, when you’re multiplying exponents and they have the same base you can set your PEMDAS aside and use this handy trick to make lighten your arithmetic load just a little.
Just Combine the Exponents and Reapply Them to the Same Base
Because an exponent is really just short hand for repeated addition, multiplying two exponential terms with the same base is really the same as just changing the exponents to something equivalent and applying them to a single instance of the base.
That sounds more complex than it really is, so let’s consider a super simple example.
Imagine you’ve encountered a problem where you’re multiplying 22 by two 23. If you worked this out long-hand using your order of operations rules, you’d break the exponents down to multiplying 2×2 and 2×2×2. That’s going to be 4×8 or 32. But, it’s worth pointing out that’s also exactly 2×2×2×2×2 or 25. That’s not a coincidence if you think about it, and generally you can regard this problem like this…
22 × 23 = 2(2+3) = 25 = 2×2×2×2×2 = 32
In a more general algebraic form, that looks like this…
nx × ny = n( x + y )
This is more than just a great shortcut when working arithmetic with actual numbers… When you’re manipulating equations in algebra, this is a tool you can use to whittle down the complexity of equations you’re manipulating.
Bonus: This Exponent Trick Works in Reverse, Too!
While we always prefer to make things less complicated where we can, sometimes algebra requires us to take a step backward before we can go forward with something else. You can use this combining exponents trick in reverse to take a term and factor it out into parts, often so that you can cancel own of them away. Consider this example…
