L1 · Exponent laws I: products, quotients, powers of powers (integer bounds)

Learning goal

“I can simplify products, quotients and powers of powers using the exponent laws, justify each law by expanding into repeated multiplication, and read parentheses correctly when the base is negative.”

Base — the number or variable being repeatedly multiplied: in (-3)⁴ the base is -3; in -3⁴ the base is 3 and the negative sign is applied afterwards.

Exponent (power) — the small raised number counting the repeated factors: 2⁵ = 2×2×2×2×2.

Product of powers law — a^m × a^n = a^(m+n): same base, multiply -> ADD the exponents.

Power of a power law — (a^m)^n = a^(mn): a power raised to a power -> MULTIPLY the exponents. (The quotient of powers law, a^m ÷ a^n = a^(m-n), is introduced in Chunk B.)

Worked example

Simplify 2³ × 2² by expansion. Step 1: Expand each power: 2³ = 2×2×2 and 2² = 2×2. Step 2: Join them: (2×2×2) × (2×2) - that is five 2s multiplied: 2⁵. Step 3: Count: 3 factors + 2 factors = 5 factors. So 2³ × 2² = 2^(3+2) = 2⁵ = 32. Say it plainly: multiplying powers with the SAME base adds the exponents, because you are just joining two strings of the same factor. Then the parentheses investigation: the base of (-2)⁴ is -2 (the parentheses include the sign), so (-2)⁴ = (-2)(-2)(-2)(-2) = 16. In -2⁴ the base is 2 and the minus is applied last: -2⁴ = -(2×2×2×2) = -16. Same digits, different base, different answer.

Check yourself — explain it

Why does 2³ × 2² = 2⁵ and not 2⁶?

Paired problem — example still visible

Simplify x⁴ · x³ and evaluate (-3)² and -3² -> x⁷; (-3)² = 9; -3² = -9.

Faded completion — last step is yours

"3² × 3³ = 3^(2+___) = 3^___ = ______" -> 3; 5; 243

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

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