7.1 Operations with Exponents & Simplifying Algebraic Expressions

7.1 Operations with Exponents & Simplifying Algebraic Expressions

Quick Answer: GRE exponent rules: a^m * a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn), (ab)^n = a^n b^n, a^0 = 1 (a != 0), a^(-n) = 1/a^n. Factor difference of squares a^2 - b^2 = (a+b)(a-b), perfect square trinomial a^2 + 2ab + b^2 = (a+b)^2, and general quadratics by the ac-method.

Why Exponents Matter on the GRE

ETS lists "operations with exponents" first among algebra topics. Expect 3-5 exponent questions per Quant section, often mixed with simplifying algebraic expressions. Mastery of the six core rules plus the four common factoring patterns eliminates most algebra errors.

The Six Exponent Rules (Memorize)

Common Exponent Traps

The GRE exploits these misconceptions repeatedly:

1. Different bases don't add: 2^3 * 3^2 is NOT 6^5. Convert to a common base or compute numerically.
2. Exponents don't distribute over addition: (a + b)^2 is NOT a^2 + b^2. The correct expansion is (a + b)^2 = a^2 + 2ab + b^2.
3. Negative exponents don't make negatives: x^(-2) = 1/x^2, NOT -x^2. The negative exponent means "reciprocal."
4. 0^0 is undefined on the GRE. Don't assume any value.

Worked Example 1: Simplify with Multiple Rules

Simplify (2x^3 y^(-2))^2 / (4x^2 y^3).

• Step 1 - Apply power of a power on the numerator: (2x^3 y^(-2))^2 = 2^2 * (x^3)^2 * (y^(-2))^2 = 4x^6 y^(-4).
• Step 2 - Write the full fraction with positive exponents: 4x^6 / (4x^2 y^3 y^4) using y^(-4) = 1/y^4.
• Step 3 - Combine y terms in the denominator: y^3 * y^4 = y^7.
• Step 4 - Simplify coefficients and x: 4x^6 / (4x^2 y^7) = x^4 / y^7.

Answer: x^4 / y^7.

Factoring and Simplifying Algebraic Expressions

ETS lists "factoring and simplifying algebraic expressions" as the second algebra topic. The GRE rewards pattern recognition: spot the factoring form, apply the formula, and reach the answer faster than grinding through expansions.

Greatest Common Factor (GCF)

Always pull the GCF first. For 6x^3 - 12x^2 + 18x, the GCF is 6x, giving 6x(x^2 - 2x + 3).

Difference of Squares

a^2 - b^2 = (a + b)(a - b). Example: 4x^2 - 9 = (2x + 3)(2x - 3).

Perfect Square Trinomials

• a^2 + 2ab + b^2 = (a + b)^2
• a^2 - 2ab + b^2 = (a - b)^2

Example: x^2 + 6x + 9 = (x + 3)^2.

General Quadratic Factoring (ac-Method)

For ax^2 + bx + c, find two numbers that multiply to ac and add to b. Then split the middle term and factor by grouping.

Worked Example 2: Factor 2x^2 + 7x + 3.

• ac = 6; need two numbers multiplying to 6 and adding to 7 -> 6 and 1.
• Split: 2x^2 + 6x + 1x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

Simplifying Rational Expressions

Factor numerator and denominator, then cancel common factors - but only factors, never terms.

Worked Example 3: Simplify (x^2 - 9) / (x^2 + 6x + 9).

• Numerator: x^2 - 9 = (x + 3)(x - 3).
• Denominator: x^2 + 6x + 9 = (x + 3)^2.
• Cancel (x + 3): result is (x - 3) / (x + 3). Note x != -3 (the original expression is undefined there).

Fractional (Rational) Exponents

The GRE occasionally uses fractional exponents, which combine powers and roots:

• a^(1/n) = the nth root of a. Example: 8^(1/3) = 2.
• a^(m/n) = (nth root of a)^m = nth root of (a^m). Example: 27^(2/3) = (cube root of 27)^2 = 3^2 = 9.

Treat the numerator of the fraction as the power and the denominator as the root. Fractional exponents follow the same six rules as integer exponents, so x^(1/2) * x^(1/2) = x^1 = x (a useful check for sqrt(x) squared).

Worked Example 4: Simplify (8x^6)^(2/3).

• Rewrite using cube root then square: (8x^6)^(2/3) = ((8x^6)^(1/3))^2.
• Cube root: (8)^(1/3) = 2 and (x^6)^(1/3) = x^2, giving (2x^2)^2.
• Square: 4x^4.

Strategic Tips for the GRE

• Match bases before computing. If 8^x = 16, rewrite as 2^(3x) = 2^4, so 3x = 4 and x = 4/3.
• Factor first, expand second. Factored form reveals roots; expanded form obscures them.
• Watch for caveats like "x != 0" or "a is positive" in question stems - they determine which exponent rules apply.
• Plug in numbers when algebra stalls. If x^(-2) is unclear, try x = 2: 2^(-2) = 1/4.
• For QC exponent questions, test a fraction between 0 and 1 (like 1/2) and a number greater than 1 - exponents behave differently for bases under versus over 1.