Grade 10 — Mathematics 10C › C10U05 Exponents

Grade 10 › C10U05 Exponents › L2
GRADE 10 · ALBERTAOUTCOME AN3

L2 · Exponent Laws II: Powers of Powers; the Zero Exponent

Learning goal

“I can raise a power to a power by multiplying exponents, apply a power to every factor of a product or quotient, explain why any nonzero base to the exponent 0 equals 1, and tell these laws apart from the L1 laws.”

Power of a power — a power used as the base of another power, written (a^m)^n. The brackets say: the whole power a^m is the thing being repeatedly multiplied.
Power of a product — an exponent sitting outside a product, (ab)^m: it applies to every factor inside.
Power of a quotient — an exponent sitting outside a quotient, (a/b)^n: it applies to the numerator and the denominator separately.
Zero exponent — an exponent of 0. Its value is the surprise of this lesson: a^0 = 1 for every a != 0.
Worked example

Simplify (2^3)^2, and evaluate. Step 1: Read the brackets: the base of the outer power is the whole power 2^3. So (2^3)^2 means 2^3 taken as a factor, twice: (2*2*2)(2*2*2). Step 2: Count the factors of 2: two groups of 3, which is 3 * 2 = 6 factors. So (2^3)^2 = 2^6 = 64. Check by values: 2^3 = 8, and 8^2 = 64. The law, stated: to raise a power to a power, keep the base and multiply the exponents - (a^m)^n = a^(mn). Contrast with L1: 2^3 * 2^2 joins two strings (counts add); (2^3)^2 repeats one string (counts multiply). The brackets are the difference. Second example - the zero exponent, by pattern: read the powers of 3 downward: 3^4 = 81, 3^3 = 27, 3^2 = 9, 3^1 = 3. Each step down divides the value by 3. Continue the pattern one more step: 3^0 = 3 / 3 = 1. Same answer from L1: the quotient law gives 3^2 / 3^2 = 3^(2-2) = 3^0, and 3^2 / 3^2 is plainly 1. Two routes, one value: a^0 = 1, for a != 0. At base 0 the pattern collapses - stepping down from 0^1 = 0 would mean dividing by 0 - so 0^0 is given no value at all.

Check yourself — explain it

Why does (2^3)^2 multiply the exponents while 2^3 * 2^2 adds them?

Paired problem — example still visible

Simplify (x^2)^5 -> x^10 (five repeats of a 2-factor string: 2 * 5 = 10 factors)

Faded completion — last step is yours

"(3^2)^3 = 3^(2 * ___) = 3^___" -> 3; 6

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz