Same base multiplication

When the bases match, add the exponents. Example: 2^3 x 2^4 = 2^(3+4) = 2^7 = 128. Do not multiply the exponents here; you only add them when multiplying same bases.

Same base division

When dividing powers with the same base, subtract the exponents. Example: 5^6 / 5^2 = 5^4 = 625. This is faster than writing out all the factors.

Power of a power

When a power is raised to another power, multiply the exponents. Example: (3^2)^4 = 3^8 = 6561. This is the case where exponents multiply.

Zero and negative exponents

Any nonzero number to the 0 power is 1. Example: 9^0 = 1. A negative exponent means reciprocal. Example: 2^-3 = 1/2^3 = 1/8.

Powers of 2 worth memorizing

Know the common powers cold: 2^5 = 32, 2^6 = 64, 2^7 = 128, 2^8 = 256, 2^9 = 512, and 2^10 = 1024. These show up often in mental math, roots, logs, and base conversion style problems.

Practice problems

  1. 2^3 x 2^4
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    Same base, so add exponents. 2^7 = 128.

  2. 5^6 / 5^2
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    Same base, so subtract exponents. 5^4 = 625.

  3. (3^2)^4
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    Power of a power, so multiply exponents. 3^8 = 6561.

References

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