Egypt Preparatory 3 Mathematics, Week 1: Power of a power, of a product and of a quotient: the three laws and the sign of an even or odd power

Powers of powers and exponential equations

Week 1 · Powers of powers and exponential equations: the laws of exponents for real bases and integer exponents, and equations of the form a^x = b · 3 lessons

Power of a power, of a product and of a quotient: the three laws and the sign of an even or odd power

Mathematics Prep 3 Term 1, Unit 1, Lessons 1–2

GoalUse the laws (aᵐ)ⁿ = aᵐⁿ, (aᵐ × bᵏ)ⁿ and (aᵐ ÷ bᵏ)ⁿ, and decide the sign of a power of a negative number.

New words

power of a power · a power whose base is itself a power, such as (2³)²base · the number that is multiplied by itself, the 2 in 2⁵exponent · how many times the base is used as a factor, the 5 in 2⁵even and odd exponent · a negative base gives a positive power for an even exponent and a negative one for an odd exponent

Explanation

(2³)² means two copies of 2³ multiplied: 2³ × 2³ = 2⁶. So to raise a power to a power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. It is not aᵐ⁺ⁿ and it is not “a to the power mⁿ”: (2³)² = 64 but 2 to the power 3² is 2⁹ = 512. The same idea works for a product and for a quotient. Every factor inside the bracket gets the outside exponent: (aᵐ × bᵏ)ⁿ = aᵐⁿ × bᵏⁿ and (aᵐ ÷ bᵏ)ⁿ = aᵐⁿ ÷ bᵏⁿ, with b ≠ 0. Finally the sign: (−a)ⁿ = aⁿ when n is even, and −aⁿ when n is odd. Watch the bracket: (−2)⁴ = 16 but −2⁴ = −16, because in −2⁴ only the 2 is raised to the power and the minus sign is applied afterwards. A sum is different: (a + b)ⁿ is not aⁿ + bⁿ.

Law	Example	Value
(aᵐ)ⁿ = aᵐⁿ	(2³)² = 2⁶	64
(aᵐ × bᵏ)ⁿ = aᵐⁿ × bᵏⁿ	(2² × 3)³ = 2⁶ × 3³	1728
(aᵐ ÷ bᵏ)ⁿ = aᵐⁿ ÷ bᵏⁿ	(3² ÷ 2)² = 3⁴ ÷ 2²	81 ÷ 4
(−a)ⁿ	(−2)³ , (−2)⁴	−8 , 16

Worked examples

Simplify (5²)³ and say why it is not 5⁵. Multiply the exponents: (5²)³ = 5⁶ = 15 625, because 2 × 3 = 6. Adding would give 5⁵, which is the product 5² × 5³, a different thing. Check: 5² = 25 and 25 × 25 × 25 = 15 625.

Step	Factor	Becomes
number	2	2⁴ = 16
letter x	x³	x¹²
letter y	y	y⁴

Simplify (2x³y)⁴ and (3a² ÷ b³)² (b ≠ 0). Give 4 to every factor: (2x³y)⁴ = 2⁴ × x¹² × y⁴ = 16x¹²y⁴. Give 2 to every factor: (3a² ÷ b³)² = 9a⁴ ÷ b⁶.

Find the signs: (−3)³, (−x²)³ and (−x²)². (−3)³ = −27 (odd exponent). (−x²)³ = −x⁶ (odd). (−x²)² = x⁴ (even). Compare −3² = −9, where the 3 alone is squared.

Class activity

Calculator check. Use your calculator to find (2³)² and then 2 to the power 3² (that is 2⁹). Why are the answers different? Then try (−2)⁴ and −2⁴ and write what the brackets do.

Practice

Write (7²)⁴ as one power of 7.

Simplify (3x²y)³.

Simplify (2a³ ÷ b²)⁴ (b ≠ 0).

Find (−2)⁵ and −2⁴.

Which is larger, 2⁶ or 3⁴? Write both with the exponent 2.