LAWS OF ALGEBRA

QUESTION 1 [4 marks] — Criterion A Medium

Simplify each expression:

a. $x^5 \times x^3$

[1]

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$$x^8$$

b. $\dfrac{y^9}{y^4}$

[1]

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$$y^5$$

c. $(a^3)^4$

[1]

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$$a^{12}$$

d. $\dfrac{m^6 \times m^2}{m^3}$

[1]

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$$m^5$$

QUESTION 2 [3 marks] — Criterion A Medium

Simplify $\dfrac{p^4 \times p^7}{p^5 \times p^2}$, showing each index law used.

a. Simplify the numerator and denominator separately first.

[2]

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Numerator: $p^{4+7}=p^{11}$. Denominator: $p^{5+2}=p^7$.

b. Complete the simplification.

[1]

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$$\frac{p^{11}}{p^7}=p^4$$

QUESTION 3 [4 marks] — Criterion B Medium

Investigate what happens when you raise a power to another power, e.g. $(x^2)^3$.

a. Expand $(x^2)^3$ fully (writing out all factors) to find the simplified result.

[2]

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$(x^2)^3=x^2\times x^2\times x^2 = x^{2+2+2}=x^6$

b. Repeat for $(x^3)^2$ and $(x^4)^2$. State the general rule connecting the two exponents to the final exponent.

[2]

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$(x^3)^2=x^6$. $(x^4)^2=x^8$. Rule: $(x^m)^n = x^{mn}$ — multiply the exponents.

QUESTION 4 [3 marks] — Criterion C Medium

A student simplifies $x^5 \times x^3$ as $x^{15}$ (multiplying the exponents instead of adding).

a. Explain the error, and verify the correct answer using $x=2$.

[3]

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The student confused the multiplication rule ($x^m\times x^n=x^{m+n}$, ADD exponents) with the power-of-a-power rule ($(x^m)^n=x^{mn}$, MULTIPLY exponents) — these are different situations. Correct: $x^5\times x^3=x^8$. Check with $x=2$: $2^5\times2^3=32\times8=256$, and $2^8=256$ ? (while $2^{15}=32768$, clearly different).

QUESTION 5 [3 marks] — Criterion D Medium

A cube-shaped storage container has side length $x^2$ cm.

a. Write and simplify an expression for the volume of the cube.

[2]

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$$V=(x^2)^3=x^6 \text{ cm}^3$$

b. If $x=2$, find the numerical volume.

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$$V=2^6=64 \text{ cm}^3$$

QUESTION 6 [7 marks] — Criterion A Medium

Simplify each expression using the index laws:

a. $\dfrac{x^8 \times x^3}{x^5}$

[2]

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$$\frac{x^{11}}{x^5}=x^6$$

b. $(x^4)^2 \div x^3$

[2]

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$$x^8 \div x^3 = x^5$$

c. $\dfrac{(x^3)^2 \times x}{x^4}$

[3]

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$$\frac{x^6\times x}{x^4}=\frac{x^7}{x^4}=x^3$$

QUESTION 7 [5 marks] — Criterion A Hard

A cell culture's population model uses $P(t)=P_0 \times 2^{t/3}$, where $P_0$ is the initial population and $t$ is time in hours (the population doubles every 3 hours).

a. If $P_0=500$, find the population after 9 hours.

[2]

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$$P(9)=500\times2^{9/3}=500\times2^3=500\times8=4000$$

b. Find, using index laws, an expression for the RATIO $\dfrac{P(12)}{P(6)}$ in simplest index form, and explain what this ratio represents physically.

[3]

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$$\frac{P(12)}{P(6)} = \frac{P_0\times2^{12/3}}{P_0\times2^{6/3}} = \frac{2^4}{2^2}=2^2=4$$ This represents how many times LARGER the population is at 12 hours compared to 6 hours — since 12 hours is 6 hours (2 doubling periods) later than 6 hours, the population has doubled twice, i.e. quadrupled.

QUESTION 8 [6 marks] — Criterion B Hard

Investigate the pattern in $x^n \div x^n$ for different values of $n$, connecting it to the zero index law.

a. Using the DIVISION index law ($x^m\div x^n=x^{m-n}$), simplify $x^5\div x^5$, $x^{10}\div x^{10}$, and $x^1\div x^1$.

[3]

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$x^5\div x^5=x^0$. $x^{10}\div x^{10}=x^0$. $x^1\div x^1=x^0$ — all give $x^0$, regardless of the exponent value.

b. Since ANY nonzero number divided by itself directly equals 1, and the index law gives $x^0$ for all these cases, what must $x^0$ always equal? Explain why this reasoning works for ANY exponent $n$, not just the specific ones tested.

[3]

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$x^0$ must always equal 1 (for $x\ne0$), since $x^n\div x^n$ is ALWAYS exactly 1 by direct calculation (anything divided by itself is 1), regardless of what $n$ is — and the index law ALWAYS gives $x^{n-n}=x^0$ for this same calculation. Since both methods calculate the identical quantity $x^n\div x^n$ for ANY $n$, they must always agree, proving $x^0=1$ universally (for nonzero $x$), not just for the specific cases tested.

QUESTION 9 [3 marks] — Criterion C Medium

A student simplifies $\dfrac{x^9}{x^3}$ as $x^3$ (dividing the exponents instead of subtracting them).

a. Explain the error, and verify the correct answer using $x=2$.

[3]

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The division index law requires SUBTRACTING exponents ($x^m\div x^n=x^{m-n}$), not dividing them. Correct: $x^9\div x^3=x^6$. Verify with $x=2$: $2^9\div2^3=512\div8=64$, and $2^6=64$ ? (while the student's $2^3=8\ne64$).

QUESTION 10 [6 marks] — Criterion D Hard

A data centre's storage doubles capacity every time a new server rack is added, following $S(n)=1024\times2^n$ GB, where $n$ is the number of additional racks installed beyond the base configuration.

a. Find the storage capacity after adding 5 additional racks.

[2]

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$$S(5)=1024\times2^5=1024\times32=32768 \text{ GB}$$

b. The company needs at least 500,000 GB of storage. Using index laws (not trial and error), find the minimum number of additional racks needed, and discuss ONE practical limitation of this doubling model that would make it unrealistic to keep adding racks indefinitely (e.g. physical space, cost, power requirements).

[4]

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Need $1024\times2^n\ge500000 \Rightarrow 2^n\ge488.3 \Rightarrow n\ge9$ (since $2^8=256<488.3$ but $2^9=512\ge488.3$). So 9 additional racks are needed. Limitation: doubling capacity with every added rack assumes UNLIMITED physical space, power supply, and cooling capacity — in reality, data centres have finite floor space and power infrastructure, meaning this exponential doubling model would eventually hit a hard physical or financial limit long before storage needs could keep scaling this way indefinitely.

Index laws