Year 8 Mathematics | Victorian Curriculum 2.0
Exponent laws
Topic 02 | Number & Algebra | Answer key

Year 8 core - answers

Fluency

Apply one law at a time

    1. 29=5122^9 = 512
    2. a7a^7
    3. x6x^6
    4. m5m^5
    5. 36=7293^6 = 729
    6. p20p^{20}
    7. 11
    8. 11
    9. 11 (by quotient rule the exponent is 00)
    10. x3y6x^3 y^6. Method: raise each factor to the power 33.
Fluency

Combine the laws

    1. a4a^4. Method: top a8a^8; a8÷a4=a4a^8 \div a^4 = a^4.
    2. bb. Method: top b6b^6; b6÷b5=b1=bb^6 \div b^5 = b^1 = b.
    3. 8m58 m^5. Method: (2m)3=8m3(2m)^3 = 8 m^3; ×m2=8m5\times m^2 = 8 m^5.
    4. 5x35 x^3.
    5. 10a710 a^7.
    6. 27x627 x^6.
    7. a3b4a^3 b^4.
    8. x4x^4. Method: top x12⋅x2=x14x^{12} \cdot x^2 = x^{14}; x14÷x10=x4x^{14} \div x^{10} = x^4.
Reasoning

Explain and spot the mistake

    1. Wrong. 23×23=23+3=26=642^3 \times 2^3 = 2^{3 + 3} = 2^6 = 64. Sam added 3×33 \times 3 or used the wrong law.
    2. Dividing anything non-zero by itself is 11. The quotient rule says anan=an−n=a0\dfrac{a^n}{a^n} = a^{n-n} = a^0. So a0=1a^0 = 1.
    3. Wrong. (a2)3=a2×3=a6(a^2)^3 = a^{2 \times 3} = a^6 (power of a power multiplies, not adds, the exponents).
    4. Not formally Year 8 (appears in Year 9). Using the quotient rule consistently: x3x5=x−2\dfrac{x^3}{x^5} = x^{-2}, which means 1x2\dfrac{1}{x^2}.
Problem solving

Applications

    1. 2n2^n cells; after 88 hours: 28=2562^8 = 256 cells.
    2. Volume becomes (2s)3=8s3(2s)^3 = 8 s^3, so 88 times larger.
    3. 2102^{10} times bigger. (220÷210=210=10242^{20} \div 2^{10} = 2^{10} = 1024.)
    4. 100n2100 n^2 cm², or 102n210^2 n^2.

Challenge - answers

Reasoning

Harder reasoning

    1. 4x5y34 x^5 y^3. Method: top =4x6y2×x3y6=4x9y8= 4 x^6 y^2 \times x^3 y^6 = 4 x^9 y^8; divide by x4y5x^4 y^5 → 4x5y34 x^5 y^3.
    2. am+n=am×an=8×4=32a^{m+n} = a^m \times a^n = 8 \times 4 = 32.
    3. P=5000×26=5000×64=320 000P = 5000 \times 2^6 = 5000 \times 64 = 320\,000.
    4. 2112^{11}. Method: 8=238 = 2^3, 4=224 = 2^2, 32=2532 = 2^5. Expression becomes 215×26210=221210=211\dfrac{2^{15} \times 2^6}{2^{10}} = \dfrac{2^{21}}{2^{10}} = 2^{11}.
Year 8 Mathematics study companion | Answer key