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Solve the following: i) State the five (5) Laws of Indices ii) Simplify 2a + 5a x 3a - a

Mathematics

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Q1. Solve the following:

  • i) State the five (5) Laws of Indices

    1. Product Rule: am×an=am+na^m \times a^n = a^{m+n}
    2. Quotient Rule: aman=am−n\frac{a^m}{a^n} = a^{m-n}
    3. Power Rule: (am)n=amn(a^m)^n = a^{mn}
    4. Zero Exponent Rule: a0=1a^0 = 1 (where a≠0a \neq 0)
    5. Negative Exponent Rule: a−n=1ana^{-n} = \frac{1}{a^n} (where a≠0a \neq 0)

  • ii) Simplify 2a+5a×3a−a2a + 5a \times 3a - a

  1. Perform the multiplication first.

    $$
5a \times 3a = 15a^2
$$

  2. Substitute this back into the expression.

    $$
2a + 15a^2 - a
$$

  3. Combine the like terms (2a2a and −a-a).

    $$
15a^2 + (2a - a) = 15a^2 + a
$$
The simplified expression is $\boxed{15a^2 + a}$.

    Q2. Simplify the following:

    • i) a3b2c4abc−2\frac{a^3 b^2 c^4}{abc^{-2}} and evaluate when a=3,b=18,c=2a=3, b=\frac{1}{8}, c=2

  4. Simplify the expression using the laws of indices.

    For $a$: $\frac{a^3}{a^1} = a^{3-1} = a^2$  
For $b$: $\frac{b^2}{b^1} = b^{2-1} = b^1 = b$  
For $c$: $\frac{c^4}{c^{-2}} = c^{4 - (-2)} = c^{4+2} = c^6$  
The simplified expression is $a^2 b c^6$.

  5. Substitute the given values a=3,b=18,c=2a=3, b=\frac{1}{8}, c=2 into the simplified expression.

    $$
(3)^2 \times \left(\frac{1}{8}\right) \times (2)^6
$$

  6. Calculate the powers.

    $$
3^2 = 9
$$
$$
2^6 = 64
$$

  7. Perform the multiplication.

    $$
9 \times \frac{1}{8} \times 64
$$
$$
9 \times \frac{64}{8}
$$
$$
9 \times 8
$$
$$
72
$$
The evaluated value is $\boxed{72}$.

    Q3. A deposit of N50 is made on the first day of each month in a saving account that pays 6% compounded monthly. What is the balance at the end of 2 years?

    This is an annuity due problem.

    • Monthly Payment (PMT) = N50
    • Annual Interest Rate (r) = 6% = 0.06
    • Number of compounding periods per year (n) = 12 (monthly)
    • Time (t) = 2 years
    • Total number of payments (N) = n×t=12×2=24n \times t = 12 \times 2 = 24
    • Monthly Interest Rate (i) = rn=0.0612=0.005\frac{r}{n} = \frac{0.06}{12} = 0.005

    The formula for the future value of an annuity due is:

    FVdue=PMT×((1+i)N−1)i×(1+i)FV_{due} = PMT \times \frac{((1+i)^N - 1)}{i} \times (1+i)

  8. Calculate (1+i)N(1+i)^N.

    1+i=1+0.005=1.0051+i = 1 + 0.005 = 1.005 (1.005)24≈1.127159776(1.005)^{24} \approx 1.127159776

  9. Substitute the values into the formula.

    FVdue=50×(1.127159776−1)0.005×(1.005)FV_{due} = 50 \times \frac{(1.127159776 - 1)}{0.005} \times (1.005) FVdue=50×0.1271597760.005×1.005FV_{due} = 50 \times \frac{0.127159776}{0.005} \times 1.005 FVdue=50×25.4319552×1.005FV_{due} = 50 \times 25.4319552 \times 1.005 FVdue=1271.59776×1.005FV_{due} = 1271.59776 \times 1.005 FVdue≈1277.9557FV_{due} \approx 1277.9557

  10. Round to two decimal places for currency.

    The balance at the end of 2 years is N1277.96\boxed{\text{N}1277.96}.

    Q4. A painter is paid N4.20 per hour for a basic 36 hour week, and overtime is paid at one and a third times this rate. Determine how many hours the painter has to work in a week to earn N212.80.

  11. Calculate the earnings for the basic 36 hours.

    Basic hourly rate = N4.20
    Basic earnings = 36 hours×N4.20/hour=N151.2036 \text{ hours} \times \text{N}4.20/\text{hour} = \text{N}151.20

  12. Calculate the remaining amount the painter needs to earn.

    Total desired earnings = N212.80
    Remaining earnings = N212.80 - N151.20 = N61.60

  13. Calculate the overtime hourly rate.

    Overtime rate factor = 113=431 \frac{1}{3} = \frac{4}{3}
    Overtime hourly rate = 43×N4.20=N5.60/hour\frac{4}{3} \times \text{N}4.20 = \text{N}5.60/\text{hour}

  14. Calculate the number of overtime hours required to earn the remaining amount.

    Overtime hours = Remaining earningsOvertime hourly rate=N61.60N5.60/hour=11 hours\frac{\text{Remaining earnings}}{\text{Overtime hourly rate}} = \frac{\text{N}61.60}{\text{N}5.60/\text{hour}} = 11 \text{ hours}

  15. Calculate the total hours the painter has to work.

    Total hours = Basic hours + Overtime hours = 36 hours+11 hours=47 hours36 \text{ hours} + 11 \text{ hours} = 47 \text{ hours}
    The painter has to work a total of 47 hours\boxed{47 \text{ hours}}.

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