Exponents tell you how many times to multiply the base number – that is, the number the exponent is "stuck to" – by itself. So for example if you have 62 that tells you to multiply 6 by itself twice, or 6 × 6. If you see 35 it tells you to multiply 3 by itself 5 times, or 3 × 3 × 3 × 3 × 3. You can't arbitrarily strip exponents out of individual expressions, because that would change the value of the expression. But if they show up in a fraction or an equation, there are a couple of tricks you can use to get rid of them.
Factoring Out Exponents
The easiest way to get rid of exponents is to write out the multiplication they imply, identify any shared factors in all terms, then cancel them out. But you can't just randomly cancel any sets of exponents you come across; at least the base number has to be the same. So if you have a fraction like b4/a5, you can't actually cancel anything out, because the multiplication they imply is as follows:
(b × b × b × b) / (a × a × a × a × a)
As you can see, unless a and b happen to be equal or share some common factors between them (which you have no way of knowing), there are no common factors to be canceled. But if you have a fraction like a4/a2, you can do some cancelling.
Write Out the Multiplication
Write out the multiplication implied by the exponents. To continue the latest example, this gives you:
(a × a × a × a) / (a × a)
Cancel Similar Terms
You can cancel out (a × a) from both the numerator and the denominator of the fraction. This gives you:
(a × a) / 1
You can either leave the answer in this form or, if your teacher prefers, put it back into exponent form. The exponents aren't gone entirely, but they're simpler, and actually your new answer of:
Can be simplified to a2, with no more fraction required since the denominator was 1.
Radicals Cancel Exponents
Before you get on to the next ways of "getting rid" of or canceling exponents, take a moment to review the inverse operation for an exponent, the radical or root. These two functions are inverses because if you see one – say, an exponent – applying the inverse – in this case, the root – removes the exponent. When you first learn about exponents you usually deal with squares or the second power, which look like this: a2 (where a is any base number). In that case, the square root, usually written just as √, is the inverse operation.
But when you start dealing with exponents higher than two, you also need radicals or roots of a correspondingly higher order. For example, the inverse operation any cubed number like a3 is the cube root, 3√. The inverse operation for any number raised to the fourth power, such as a4, is the fourth root, which is written as 4√. And so on. Note that the index of the root – that is, the little number to the left of the radical sign – matches the exponent it's an inverse function for.
Apply the Equivalent Root in an Equation
Now, imagine that you have an equation with a cubed number hiding in it somewhere. We don't need to know what the number being cubed actually is; we can use a letter, like y, to mark its place. Here's how you'd solve an equation like 3 + y3 = 11.
Isolate the Exponential Expression
Use simple operations like addition, subtract, division or multiplication, applied equally to both sides of the equation, to isolate the exponential expression on one side of the equals sign. In this case, that's as simple as subtracting 3 from both sides of the equation. This gives you:
3 + y3 - 3 = 11 - 3
Which simplifies to:
y3 = 8
Apply the Appropriate Root
Next, apply the appropriate root to cancel the exponent. Again, you must do this equally to both sides of the equation. Because the exponent is 3, you must apply the third root or cube root:
3√(y3) = 3√8
It so happens that 3√8 = 2, so once you apply the cube root to both sides of the equation, you get a tidy answer:
y = 2
Watch out – there's a sneaky trap here! When you apply a root to the side of the equation where you've isolated your exponent, you must apply it to the entire other side of the equation too. Sometimes this works out easily, as in the example just given. But sometimes it isn't so pretty. Say you had been asked to solve y3 = a3 + 8. You can see that each of these numbers has a tidy cube root, so it's tempting to work 3√(y3) = 3√(a3) + 3√8 and call it good. But what you actually have to calculate is this:
3√(y3) = 3√(a3 + 8)
...with both a3 and 8 under the same root, which would give you a very different answer.