What Is Exponents and roots on the GMAT?
Exponents and roots are the mathematical shorthand for repeated multiplication and its inverse. On the GMAT, you'll need to simplify expressions with powers and radicals quickly and accurately, as they appear in many quantitative problems.
Definition
Exponents (or powers) represent how many times a number (the base) is multiplied by itself. For example, \(5^3\) means \(5 \times 5 \times 5 = 125\). Roots (or radicals) are the inverse operation: the square root of \(x\) (written \(\sqrt{x}\)) is a number that, when squared, gives \(x\). The cube root (written \(\sqrt[3]{x}\)) is the number that, when cubed, gives \(x\).
Why It Matters on the GMAT
Exponents and roots appear throughout the GMAT Quantitative section, from arithmetic to algebra to data sufficiency. You'll need to simplify expressions, solve equations, and compare values. Mastery of the rules for manipulating exponents and radicals is essential for speed and accuracy.
Concrete Example
Simplify: \(\frac{2^5 \times 2^3}{2^4}\). Using the rule \(a^m \times a^n = a^{m+n}\) and \(a^m / a^n = a^{m-n}\), we get \(2^{5+3-4} = 2^4 = 16\).
Common Mistakes
- Confusing \(a^m \times a^n\) with \((a^m)^n\). The former adds exponents, the latter multiplies them.
- Forgetting that \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), but \(\sqrt{a} + \sqrt{b}\) is NOT \(\sqrt{a+b}\).
- Misapplying the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), not a negative number.
- Overlooking that \(x^0 = 1\) for any nonzero \(x\).
How to Practice It
Drill the core rules daily: product rule, quotient rule, power of a power, negative exponents, and fractional exponents. Work through official GMAT problems that mix exponents and roots with other topics. Focus on simplifying before calculating, and always check your answer by plugging back in when possible.