Mathematics • Year 9 • Unit 1 • Lesson 11
Algebraic Expressions and Index Laws
Build fluency with index laws on variables (letters): $x^m \times x^n = x^{m+n}$, $x^m \div x^n = x^{m-n}$, $(x^m)^n = x^{mn}$. With coefficients, multiply the numbers and combine the indices. Decide whether to ADD like terms or MULTIPLY them.
1. I do — fully worked example
Read every step. Each line has a short reason on the right.
Problem. Simplify $3x^2 \times 4x^3$.
Step 1 — Spot the operation.
The operation is $\times$ — so we're MULTIPLYING terms. Coefficients $\times$, indices $+$.
Reason: $ax^m \times bx^n = (ab) x^{m+n}$.
Step 2 — Separate coefficients from variables.
$3x^2 \times 4x^3 = (3 \times 4) \times (x^2 \times x^3)$
Reason: multiplication is associative — we can pair the numbers together and the variables together.
Step 3 — Multiply the coefficients.
$3 \times 4 = 12$
Reason: just numbers — straight multiplication.
Step 4 — Apply the product rule to the variables.
$x^2 \times x^3 = x^{2+3} = x^5$
Reason: same base $x$ on both sides of the $\times$ — ADD the indices.
Step 5 — Put it together.
$3x^2 \times 4x^3 = 12 x^5$
Reason: coefficient out the front, then the variable with its simplified index.
Answer: $\mathbf{12x^5}$.
2. We do — fill in the missing steps
Same structure as Section 1, but the working is faded. Fill in each blank. 4 marks
Problem. Simplify (a) $3x^2 + 4x^2$ and (b) $3x^2 \times 4x^2$.
(a) Step 1 — Spot the operation. The operation is __________, so we are ADDING __________ terms (same variable, same index).
(a) Step 2 — Add the coefficients only:
$3x^2 + 4x^2 = (\,\_\_\, + \,\_\_\,)\,x^{\_\_\,}$
(a) Step 3 — Simplify:
$3x^2 + 4x^2 = \_\_\_\,x^{\_\_\,}$
(b) Step 1 — Spot the operation. The operation is __________, so coefficients ____________ and indices ____________.
(b) Step 2 — Multiply coefficients, add indices:
$3x^2 \times 4x^2 = (\_\_ \times \_\_)\,x^{\_\_\, + \,\_\_} = \_\_\_\,x^{\_\_\,}$
3. You do — independent practice
Show working in the space under each problem. The first four are foundation (pure-variable rules, no coefficients). The middle two are standard (coefficients with multiply/divide). The last two are extension (decide operation, decide rule).
Foundation — pure-variable rules
3.1 Simplify $x^4 \times x^3$. 1 mark
3.2 Simplify $(y^4)^2$. 1 mark
3.3 Simplify $\dfrac{a^9}{a^4}$. 1 mark
3.4 Simplify $m^6 \div m^2$. 1 mark
Standard — coefficients with multiply/divide
3.5 Simplify $5x^4 \times 2x^3$. 2 marks
3.6 Simplify $\dfrac{8 m^7}{2 m^2}$. 2 marks
Extension — decide operation, decide rule
3.7 Simplify $5x^4 + 2x^4$ and $5x^4 \times 2x^4$ separately. State which one keeps the index and why. 3 marks
3.8 Simplify $x^2 \times y^3$ (different bases). Explain in one sentence why this CAN'T be combined into a single power. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do
(a) Operation is $\mathbf{+}$ → ADDING like terms.
Step 2: $3x^2 + 4x^2 = (\mathbf{3} + \mathbf{4})\,x^{\mathbf{2}}$.
Step 3: $3x^2 + 4x^2 = \mathbf{7}\,x^{\mathbf{2}}$.
(b) Operation is $\mathbf{\times}$ → coefficients multiply and indices add.
Step 2: $3x^2 \times 4x^2 = (\mathbf{3} \times \mathbf{4})\,x^{\mathbf{2} + \mathbf{2}} = \mathbf{12}\,x^{\mathbf{4}}$.
3.1 — $x^4 \times x^3$
Product rule: $x^{4+3} = \mathbf{x^7}$.
3.2 — $(y^4)^2$
Power-of-a-power: $y^{4 \times 2} = \mathbf{y^8}$.
3.3 — $\dfrac{a^9}{a^4}$
Quotient rule: $a^{9-4} = \mathbf{a^5}$.
3.4 — $m^6 \div m^2$
Quotient rule: $m^{6-2} = \mathbf{m^4}$.
3.5 — $5x^4 \times 2x^3$
Coefficients: $5 \times 2 = 10$. Indices: $4 + 3 = 7$. Answer: $\mathbf{10x^7}$.
3.6 — $\dfrac{8 m^7}{2 m^2}$
Coefficients: $8 \div 2 = 4$. Indices: $7 - 2 = 5$. Answer: $\mathbf{4 m^5}$.
3.7 — $5x^4 + 2x^4$ vs $5x^4 \times 2x^4$
$5x^4 + 2x^4 = \mathbf{7x^4}$ — the index stays the same because we're adding like terms (only coefficients change).
$5x^4 \times 2x^4 = \mathbf{10 x^8}$ — coefficients multiply ($5 \times 2 = 10$) and indices add ($4 + 4 = 8$).
The operation ($+$ vs $\times$) decides whether the index changes.
3.8 — $x^2 \times y^3$
$x^2 \times y^3 = \mathbf{x^2 y^3}$ — nothing more we can do.
The product rule needs the SAME base on both sides of the $\times$. Here the bases are different ($x$ vs $y$), so the indices stay where they are.