Arithmetic and terms

Powers and roots

Power rules let you simplify long products in a few steps. Entrance tests check them directly and inside other tasks, such as equations and terms.

The rules

• Same base, multiply: aᵐ · aⁿ = aᵐ⁺ⁿ; divide: aᵐ / aⁿ = aᵐ⁻ⁿ.
• Power of a power: (aᵐ)ⁿ = aᵐⁿ.
• Same exponent: aⁿ · bⁿ = (ab)ⁿ.
• a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) = ⁿ√a.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Simplify and evaluate: 10⁴ · 10²

   • a1000000
   • b10100
   • c100000000
   • d100000
   Show solution

   Answer: 1000000

   Same base, product of powers: add the exponents. aᵐ · aⁿ = aᵐ⁺ⁿ.

   10⁴⁺² = 10⁶ = 1000000.

2. 2. Simplify and evaluate: 3⁴ · 3³

   • a729
   • b531441
   • c108
   • d2187
   Show solution

   Answer: 2187

   Same base, product of powers: add the exponents. aᵐ · aⁿ = aᵐ⁺ⁿ.

   3⁴⁺³ = 3⁷ = 2187.

3. 3. Evaluate: √196

   • a9
   • b98
   • c14
   • d7
   Show solution

   Answer: 14

   Pull out perfect-square factors: √(a²·k) = a·√k.

   196 = 49·4, so √196 = 7·√4 = 14.

4. 4. Evaluate: √256

   • a128
   • b8
   • c16
   • d10
   Show solution

   Answer: 16

   Pull out perfect-square factors: √(a²·k) = a·√k.

   256 = 64·4, so √256 = 8·√4 = 16.

5. 5. Evaluate: √900

   • a10
   • b30
   • c13
   • d450
   Show solution

   Answer: 30

   Pull out perfect-square factors: √(a²·k) = a·√k.

   900 = 100·9, so √900 = 10·√9 = 30.

6. 6. Simplify and evaluate: 2³ · 2²

   • a32
   • b16
   • c64
   • d12
   Show solution

   Answer: 32

   Same base, product of powers: add the exponents. aᵐ · aⁿ = aᵐ⁺ⁿ.

   2³⁺² = 2⁵ = 32.

Common mistakes

• Multiplying exponents when you should add them: a² · a³ = a⁵, not a⁶.
• Treating a sum like a product: (a + b)² is not a² + b².
• Sign errors with negative bases: (−2)² = 4 but −2² = −4.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs