How To Get Rid Of Exponents In An Algebraic Equation

Few things strike fear into the beginning algebra student like seeing exponents – expressions such as ​y​², ​x​³ or even the horrifying ​_y^x_​ – pop up in equations. In order to solve the equation, you need to somehow make those exponents go away. But in truth, that process isn't so difficult once you learn a series of simple strategies, most of which are rooted in the basic arithmetic operations you've been using for years.

Sometimes, if you're lucky, you might have exponent terms in an equation that cancel each other out. For example, consider the following equation:

\(y + 2x^2 – 5 = 2(x^2 + 2)\)

With a keen eye and a little practice, you might spot that the exponent terms actually cancel each other out, thusly:

1. Simplify Where Possible

Once you simplify the right side of the sample equation, you'll see that you have identical exponent terms on both sides of the equals sign:

\(y + 2x^2 – 5 = 2x^2 + 4\)

2. Combine/Cancel Like Terms

Subtract 2​x​² from both sides of the equation. Because you performed the same operation on both sides of the equation, you haven't altered its value. But you have effectively removed the exponent, leaving you with:

\(y – 5 = 4\)

If desired, you can finish solving the equation for ​y​ by adding 5 to both sides of the equation, giving you:

\(y = 9\)

Often problems won't be this simple, but it's still an opportunity worth looking out for.

With time, practice and lots of math classes, you'll collect formulas for factoring certain types of polynomials. It's a lot like collecting tools that you keep in a toolbox until you need them. The trick is learning to identify which polynomials can be easily factored. Here are some of the most common formulas you might use, with examples of how to apply them:

1. The Difference of Squares

If your equation contains two squared numbers with a minus sign between them – for example, ​x​² − 4² – you can factor them using the formula ​a​² − ​b​² ​= (a + b)(a − b)​. If you apply the formula to the example, the polynomial ​x​² − 4² factors to (​x​ + 4)(​x​ − 4).

The trick here is learning to recognize squared numbers even if they aren't written as exponents. For example, the example of ​x​² − 4² is more likely to be written as ​x​² − 16.

2. The Sum of Cubes

If your equation contains two cubed numbers that are added together, you can factor them using the formula

\(a^3 + b^3 = (a + b)(a^2 – ab + b^2)\)

Consider the example of ​y​³ + 2³, which you're more likely to see written as ​y​³ + 8. When you substitute ​y​ and 2 into the formula for ​a​ and ​b​ respectively, you have:

\((y + 2)(y^2 – 2y + 2^2)\)

Obviously the exponent isn't gone entirely, but sometimes this type of formula is a useful, intermediate step toward getting rid of it. For example, factoring thusly in the numerator of a fraction might create terms that you can then cancel with terms from the denominator.

3. The Difference of Cubes

If your equation contains two cubed numbers with one ​subtracted​ from the other, you can factor them using a formula very similar to that shown in the previous example. In fact, the location of the minus sign is the only difference between them, as the formula for the difference of cubes is:

\(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\)

Consider the example of ​x​³ − 5³, which would more likely be written as ​x​³ − 125. Substituting ​x​ for ​a​ and 5 for ​b​, you get:

\((x – 5)(x^2 + 5x + 5^2)\)

As before, although this doesn't eliminate the exponent entirely, it can be a useful intermediate step along the way.

If neither of the above tricks works and you have just one term containing an exponent, you can use the most common method for "getting rid of" the exponent: Isolate the exponent term on one side of the equation, and then apply the appropriate radical to both sides of the equation. Consider the example of

\(z^3 – 25 = 2\)

1. Isolate the Exponent Term

Isolate the exponent term by adding 25 to both sides of the equation. This gives you:

\(z^3 = 27\)

2. Apply the Appropriate Radical

The index of the root you apply – that is, the little superscript number before the radical sign – should be the same as the exponent you're trying to remove. So because the exponent term in the example is a cube or third power, you must apply a cube root or third root to remove it. This gives you:

\(\sqrt[3]{z^3} = \sqrt[3]{27}\)

Which in turn simplifies to:

\(z = 3\)