GRE Exponents and Roots

The rules, all of them

• Same base, multiply: x^a times x^b = x^(a+b).
• Same base, divide: x^a / x^b = x^(a-b).
• Power of a power: (x^a)^b = x^(ab).
• Distribute over multiplication: (xy)^a = x^a y^a.
• Zero and negative: x^0 = 1, and x^-a = 1 / x^a.

These apply only when the base is the same (for multiply and divide). You cannot combine 2^3 and 3^2 with exponent rules; compute them.

Fractional exponents are roots

A fractional exponent packs a root and a power together: the denominator is the root, the numerator is the power. x^(1/2) is the square root of x; x^(a/b) is the b-th root of x^a. So 8^(2/3) is the cube root of 8 (which is 2), squared, giving 4. Rewriting roots as fractional exponents lets you use all the rules above on them.

Simplifying and estimating roots

Pull perfect squares out of a radical: root 50 = root(25 x 2) = 5 root 2. You can multiply and divide roots freely (root a times root b = root(ab)) but you can only add or subtract like radicals (2 root 3 + 4 root 3 = 6 root 3). To estimate, bracket between perfect squares: root 50 sits just above 7 because 7^2 = 49.

The two traps

First, exponents do not distribute over addition: (x + y)^2 = x^2 + 2xy + y^2, never x^2 + y^2. Second, watch signs with even powers: (-2)^2 = 4, and when you solve x^2 = 9 the answer is x = 3 or x = -3, even though the root symbol itself refers to the non-negative root.

Frequently asked questions