Year 7 Mathematics | Victorian Curriculum 2.0
Squares, roots & exponents
Topic 04 | Number & Algebra | Answer key

Year 7 core - answers

Fluency

Squares and square roots

    1. 99
    2. 3636
    3. 144144
    4. 99
    5. 1212
    6. 2020
    7. Between 77 and 88
    8. Between 99 and 1010
    9. 52=255^2 = 25 is bigger than 64=8\sqrt{64} = 8
    10. 1111. Method: 5+65 + 6.
Fluency

Powers of 10 and expanded notation

    1. 10001000
    2. 100 000100\,000
    3. 3×103+4×102+2×10+73 \times 10^3 + 4 \times 10^2 + 2 \times 10 + 7
    4. 5×104+6×102+85 \times 10^4 + 6 \times 10^2 + 8
    5. 72067206
Fluency

Prime factorisation

    1. 22×52^2 \times 5
    2. 22×322^2 \times 3^2
    3. 22×3×72^2 \times 3 \times 7
    4. 22×522^2 \times 5^2
    5. 23×522^3 \times 5^2

Explain and apply - answers

Reasoning

Explain and apply

    1. Square root does not distribute over addition. Left side: 9+16=25=5\sqrt{9 + 16} = \sqrt{25} = 5. Right side: 9+16=3+4=7\sqrt{9} + \sqrt{16} = 3 + 4 = 7. Since 5≠75 \neq 7, you must add first and then take the root.
    2. 142=19614^2 = 196 and 152=22515^2 = 225. Since 200200 is only 44 above 196196 but 2525 below 225225, 200\sqrt{200} is much closer to 1414 (about 14.1414.14).
    3. 1212 m. Method: side length =144= \sqrt{144}.
    4. HCF =24= 24. Method: 72=23×3272 = 2^3 \times 3^2 and 120=23×3×5120 = 2^3 \times 3 \times 5; take the lowest power of each shared prime: 23×3=242^3 \times 3 = 24.
    5. LCM =36= 36. Method: 12=22×312 = 2^2 \times 3 and 18=2×3218 = 2 \times 3^2; take the highest power of each prime: 22×32=362^2 \times 3^2 = 36.

Extension - answers

Fluency

Using the index laws

    1. 292^9
    2. a8a^8
    3. 565^6
    4. x4x^4
    5. 363^6 (which equals 729729)
    6. m20m^{20}
    7. y6y^6. Method: subtract indices 10−410 - 4.
    8. m4m^4. Method: top m9m^9; m9÷m5=m4m^9 \div m^5 = m^4.
Reasoning

Spot the mistake (extension)

    1. Tim is wrong. 23+23=8+8=16=242^3 + 2^3 = 8 + 8 = 16 = 2^4, not 262^6. The index law for multiplying powers adds the indices, but this is addition of two equal powers - it doubles the value, increasing the index by 11 (not doubling it).
    2. Leah is wrong. The rule (am)n(a^m)^n multiplies the indices, not adds: (a2)3=a2×3=a6(a^2)^3 = a^{2 \times 3} = a^6.
    3. p3q2p^3 q^2. Method: numerator =p2+4q3+1=p6q4= p^{2 + 4} q^{3 + 1} = p^6 q^4; dividing by p3q2p^3 q^2 gives p6−3q4−2=p3q2p^{6 - 3} q^{4 - 2} = p^3 q^2.
    4. 26=642^6 = 64 cells. Method: doubling 66 times from 11 gives 1→2→4→8→16→32→641 \to 2 \to 4 \to 8 \to 16 \to 32 \to 64.
Year 7 Mathematics study companion | Answer key