Exponents and Roots: Practice Questions & Explanations

2⁵ means 2 multiplied by itself 5 times: 2 × 2 × 2 × 2 × 2 = 4 × 2 × 2 × 2 = 8 × 2 × 2 = 16 × 2 = 32. Memorize common powers: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1024. Also: 3² = 9, 3³ = 27, 3⁴ = 81; 4² = 16, 4³ = 64; 5² = 25, 5³ = 125. Exponent rules: (1) a^m · a^n = a^(m+n); (2) a^m / a^n = a^(m−n); (3) (a^m)^n = a^(m·n); (4) a⁰ = 1 (for a ≠ 0); (5) a^(−n) = 1/a^n; (6) a^(1/n) = ⁿ√a (the n-th root); (7) (ab)^n = a^n · b^n; (8) (a/b)^n = a^n / b^n. Common error: 2⁵ ≠ 2 × 5 = 10. Exponents are repeated multiplication, not multiplication.

Simplify radicals by finding perfect-square factors. Factor 48: 48 = 16 × 3, where 16 is a perfect square. So √48 = √(16 × 3) = √16 × √3 = 4√3. The perfect-square factor comes out of the radical; the remaining factor stays inside. Steps: (1) Find the largest perfect-square factor; (2) Rewrite as a product; (3) Apply √(ab) = √a × √b; (4) Simplify the perfect-square root. Perfect squares to recognize: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Example: √72 = √(36 × 2) = 6√2. Example: √200 = √(100 × 2) = 10√2. If no perfect-square factor exists beyond 1, the radical is already simplified (like √7, √15, √23). Common error: stopping with non-largest factor — √48 = √(4 × 12) = 2√12, which is not fully simplified because 12 has a perfect-square factor of 4 itself.

Negative exponents indicate reciprocals: a^(−n) = 1/a^n. So x⁻² = 1/x². With x = 4: x⁻² = 1/4² = 1/16. The negative exponent does NOT make the result negative — it makes it a reciprocal. Negative exponent rules: (1) a^(−n) = 1/a^n; (2) (a/b)^(−n) = (b/a)^n (flip the fraction, change sign); (3) 1/a^(−n) = a^n. Examples: 2⁻³ = 1/2³ = 1/8; (1/3)⁻² = 3² = 9; 10⁻⁴ = 0.0001. Zero exponent: a⁰ = 1 for any non-zero a. 0⁰ is undefined (or 1 by convention in some contexts). Fractional exponents: a^(1/n) = ⁿ√a; a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m. Example: 8^(2/3) = (³√8)² = 2² = 4. These rules combine: (1/2)⁻³ = 2³ = 8.

Multiply coefficients and apply exponent rules to variables. Coefficients: 3 × 4 = 12. Variables: x² × x³ — use the product rule a^m × a^n = a^(m+n), so x² × x³ = x^(2+3) = x⁵. Combined: 12x⁵. Common error: multiplying exponents instead of adding (multiplication of like bases ADDS exponents; raising a power to a power MULTIPLIES exponents). Distinguish: (1) x² × x³ = x⁵ (add); (2) (x²)³ = x⁶ (multiply). Other key exponent operations: (1) (3x²y)(4xy³) = 12 x³y⁴ (multiply coefficients, add exponents of each variable separately); (2) 12x⁵ ÷ 3x² = 4x³ (divide coefficients, subtract exponents); (3) (2x³)² = 4x⁶ (raise both coefficient and variable to the outside exponent); (4) (3x²)³ = 27x⁶ (3³ = 27, and (x²)³ = x⁶). Master these patterns to handle polynomial arithmetic efficiently.

When multiplying powers with the same base, add the exponents: 2³ × 2⁴ = 2^(3+4) = 2⁷. This is the product rule for exponents. Do not multiply the exponents (that rule applies to a power raised to a power) and do not change the base. So 2⁷ = 128 if you compute it out, but in simplified exponential form the answer is 2⁷. The common mistakes are adding the bases or multiplying the exponents; remember: same base, multiply the terms, add the exponents.

The square root of 144 is the number that, multiplied by itself, gives 144. Since 12 × 12 = 144, √144 = 12. Memorizing perfect squares up to at least 15² (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) makes these instant. The distractor 72 is 144 ÷ 2, a common error of halving instead of taking the root. The square root of 144 is 12.

When raising a power to another power, multiply the exponents: (3²)³ = 3^(2×3) = 3⁶. This is the power-of-a-power rule, distinct from the product rule (where you add exponents for multiplication of like bases). So (3²)³ = 3⁶, which equals 729 if computed out. The common mistake is adding the exponents (giving 3⁵) instead of multiplying them. Remember: multiply exponents when a power is raised to a power; add exponents when multiplying like bases. The simplified form is 3⁶.