Solving Algebraic Equations Involving Exponential Terms

GRE exponent problems sometimes give you an equation, and ask you to solve for either an unknown base or an unknown exponent.

Unknown Base

The key to solving algebraic expressions with an unknown base is to make use of the fact that exponents and roots are inverses, just as multiplication and division are, and so can be used to effectively cancel each other out. In the equation x³ = 8, x is raised to the third power, so to eliminate the exponent you can take the cube root of both sides of the equation.

This process also works in reverse. If you are presented with the equation you can eliminate the square root by squaring both sides. Square root and squaring cancel each other out in the same way that cube root and raising something to the third power cancel each other out. So to solve this equation, you can square both sides and get which can be simplified to x = 36.

There is one additional danger. Remember that when solving an equation where a variable has been squared, you should be on the lookout for two solutions. To solve for y in the equation y² = 100, you need to remember that y can equal either 10 or −10.

Unknown Base	Unknown Exponent
x3 = 8	2x = 8

Check Your Skills

Solve the following equations.

8. x³ = 64

9. 

10. x² = 121

Unknown Exponent

Unlike examples in the previous section, you can’t make use of the relationship between exponents and roots to help solve for the variable in the equation 2^x = 8. Instead, the key is to once again recognize that 8 is equivalent to 2³, and rewrite the equation so that you have the same base on both sides of the equal sign. If you replace 8 with its equivalent value, the equation becomes 2^x = 2³.

Now that you have the same base on both sides of the equation, there is only one way for the value of the expression on the left side of the equation to equal the value of the expression on the right side of the equation—the exponents must be equal. You can effectively ignore the bases and set the exponents equal to each other. You now know that x = 3.

By the way, when you see the expression 2^x, always call it “two TO THE xth power” or “two TO THE x.” Never call it “two x.” “Two x” is 2x, or 2 times x, which is simply a different expression. Don’t get lazy with names; that’s how you can confuse one expression for another.

The process of finding the same base on each side of the equation can be applied to more complicated exponents as well. Take a look at the equation 3^{x + 2} = 27. Once again, you must first rewrite one of the bases so that the bases are the same on both sides of the equation. Because 27 is equivalent to 3³, the equation can be rewritten as 3^{x + 2} = 3³. You can now ignore the bases (because they are the same) and set the exponents equal to each other: x + 2 = 3, which means that x = 1.

Check Your Skills

Solve for x in the following equations.

11. 2^x = 64

12. 7^{x − 2} = 49

13. 5^3x = 125