Lesson 12 ~25 min Unit 1 · Index Laws +85 XP

Simplifying with the Product Rule

Multiply algebraic terms with multiple variables, negative coefficients, and one-step expansions like $2x(3x^2 - 4x)$.

Today's hook: $3a^2 b \times 5 a^3 b^2$ — two variables, two index laws needed. What's the fastest path to the answer?

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Think First

warm-up

Predict the simplified form of $2x^2 y^3 \times 3x^4 y$. How many separate index calculations do you need? What happens to the coefficients $2$ and $3$?

Record in your workbook.

The Big Idea

+5 XP

When multiplying algebraic terms, split the job into three jobs: coefficients (multiply), each variable (apply the product rule), and signs (multiply $+/-$).

$2x^2 y^3 \times 3 x^4 y = (2 \times 3)(x^{2+4})(y^{3+1}) = 6 x^6 y^4$. Treat every variable independently — the $x$ indices add, the $y$ indices add, and the coefficients multiply. Signs follow the usual rules of multiplication.

$2x^2 y^3 \times 3x^4 y = 6 x^6 y^4$

Variable by variable

$x$ with $x$, $y$ with $y$.

Signs first

$-4a^2 \times 5a^3 = -20 a^5$.

Missing index = 1

$y$ is $y^1$.

What You'll Master

objectives

Know

Product rule for each variable
$-$ $\times$ $-$ $= +$, $-$ $\times$ $+$ $= -$
$y = y^1$ for the missing index

Understand

Why each variable's indices add separately
Why coefficients multiply but indices add
How distributive law extends to $a(b + c)$

Can Do

Simplify $2x^2 y^3 \times 3x^4 y$ to $6x^6 y^4$
Simplify $-4a^2 \times 5a^3$ to $-20a^5$
Expand $2x(3x^2 - 4x)$ to $6x^3 - 8x^2$

Words You Need

vocabulary

CoefficientNumber multiplying the variables; carries any sign.

Multi-variable terme.g. $3a^2 b$ — two letters, treated separately.

Implicit index of 1$y$ means $y^1$ for index purposes.

ExpandUse the distributive law: $a(b+c) = ab + ac$.

Sign rule$- \times - = +$, $- \times + = -$.

Independent variables$x$ and $y$ don't combine; each has its own index.

Spot the Trap

heads-up

Wrong: “$2x^2 \times 3y^3 = 6 (xy)^5$” — combining different bases.

Right: $2x^2 \times 3y^3 = 6 x^2 y^3$. Bases $x$ and $y$ stay separate.

Wrong: “$-2x^3 \times -3x^2 = -6x^5$” — forgetting two negatives make a positive.

Right: $-2x^3 \times -3x^2 = 6x^5$.

Multiple variables

+5 XP

Treat each variable independently. Make a column for each variable, add the indices in that column, then multiply the coefficients.

$3a^2 b \times 5 a^3 b^2$: coefficient column $3 \times 5 = 15$; $a$ column $2 + 3 = 5$ so $a^5$; $b$ column $1 + 2 = 3$ so $b^3$. Final: $15 a^5 b^3$.

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

Expanding before simplifying

+5 XP

When a single term multiplies a bracket, distribute first — using the product rule on each piece.

$2x(3x^2 - 4x) = 2x \cdot 3x^2 - 2x \cdot 4x = 6x^3 - 8x^2$. Coefficients multiply, indices add on each term. The two answers stay separate because $x^3$ and $x^2$ are not like terms.

$2x(3x^2 - 4x) = 6x^3 - 8x^2$

Watch Me Solve It · 3 examples

Watch Me Solve It · Two variables

+15 XP per step

PROBLEM

Simplify $2x^2 y^3 \times 3x^4 y$.

1
Group like factors
$(2 \times 3)(x^2 \times x^4)(y^3 \times y)$
2
Apply product rule to each variable
$6 \times x^{2+4} \times y^{3+1} = 6 x^6 y^4$
$y$ on its own is $y^1$.
3
Final
$6 x^6 y^4$

Answer$6 x^6 y^4$

Watch Me Solve It · Negative coefficients

+15 XP per step

PROBLEM

Simplify $-4a^2 \times 5a^3$ and $-2x^3 \times -3x^2$.

1
Decide the signs
$(-) \times (+) = (-)$; $(-) \times (-) = (+)$
2
Multiply coefficients with sign
$-4 \times 5 = -20$; $-2 \times -3 = 6$
3
Add indices
$-20 a^{2+3} = -20 a^5$; $6 x^{3+2} = 6 x^5$

Answer$-20 a^5$ and $6 x^5$

Watch Me Solve It · Expand then simplify

+15 XP per step

PROBLEM

Expand and simplify $2x(3x^2 - 4x)$.

1
Distribute
$2x(3x^2) - 2x(4x)$
2
Apply product rule to each piece
$6 x^{1+2} - 8 x^{1+1} = 6x^3 - 8x^2$
3
Check for like terms
$6x^3$ and $8x^2$ are NOT like terms
Different indices — leave as is.

Answer$6x^3 - 8x^2$

Common Pitfalls

heads-up

Forgetting to multiply coefficients

$3a^2 \times 2a = 5a^3$ is wrong — the student added the coefficients.

Fix: When multiplying, $3 \times 2 = 6$, so $6a^3$.

Combining different bases

$x^2 \times y^3 \ne (xy)^5$. Different bases can't be merged.

Fix: Make a column per variable; combine only within the column.

Sign slips

$-2 \times -3$ should be $+6$, not $-6$.

Fix: Determine the sign FIRST, then handle numbers and variables.

Copy Into Your Books

Three columns

Coefficient column $\times$
Each variable column $+$
Sign column $+/-$

Worked example

$2x^2y^3 \times 3x^4y$
$= 6 x^6 y^4$

Signs

$-4a^2 \times 5a^3 = -20a^5$
$-2x^3 \times -3x^2 = 6x^5$

Expansion

$2x(3x^2 - 4x)$
$= 6x^3 - 8x^2$
Apply product rule each piece

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Product rule drills

4 problems

Multi-variable products, sign work and a quick expansion.

1 Simplify $4 a^3 b \times 2 a b^4$.
Coefficients $4 \times 2 = 8$; $a$ indices $3 + 1 = 4$; $b$ indices $1 + 4 = 5$.$8 a^4 b^5$
2 Simplify $-3 p^2 \times 4 p^5$.
Sign $-$; coefficients $3 \times 4 = 12$; indices $2 + 5 = 7$.$-12 p^7$
3 Simplify $-5 m \times -6 m^4$.
Sign $+$; coefficients $5 \times 6 = 30$; indices $1 + 4 = 5$.$30 m^5$
4 Expand and simplify $3y(2y^2 + 5)$.
$3y \cdot 2y^2 + 3y \cdot 5 = 6y^3 + 15y$.$6 y^3 + 15 y$

Complete in your workbook.

Quick Check · 5 questions

Simplify $2x^2 y^3 \times 3x^4 y$.

+10 XP

Simplify $-4a^2 \times 5a^3$.

+10 XP

Simplify $-2x^3 \times -3x^2$.

+10 XP

Expand $2x(3x^2 - 4x)$.

+10 XP

Simplify $3a^2 b \times 5 a^3 b^2$.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Simplify each, showing coefficient and variable steps: (a) $5x^3 \times 4x^2$, (b) $2 a^2 b \times 7 a^3 b^4$, (c) $-3 m^4 n \times 2 m n^2$.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Expand and simplify: (a) $3x(2x^2 + 5x)$, (b) $-2a(3a^3 - 4)$, (c) $4y(y^2 - 2y + 1)$.

Answer in your workbook.

ReasonHard3 MARKS

Q8. The area of a rectangle is given by $A = (2x^2 y)(5x y^3)$. Write the area as a single simplified term, and explain why the variables $x$ and $y$ keep separate indices in your answer.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. D — $6x^6 y^4$.

2. B — $-20a^5$.

3. A — $6x^5$.

4. C — $6x^3 - 8x^2$.

5. D — $15 a^5 b^3$.

Show Your Working Model Answers

Q6 (3 marks): (a) $5 \times 4 = 20$, $x^{3+2} = x^5$, so $20x^5$ [1]; (b) $2 \times 7 = 14$, $a^{2+3} = a^5$, $b^{1+4} = b^5$, so $14 a^5 b^5$ [1]; (c) sign $-$, $3 \times 2 = 6$, $m^{4+1} = m^5$, $n^{1+2} = n^3$, so $-6 m^5 n^3$ [1].

Q7 (3 marks): (a) $6x^3 + 15x^2$ [1]; (b) $-6 a^4 + 8 a$ [1]; (c) $4y^3 - 8y^2 + 4y$ [1].

Q8 (3 marks): $A = (2 \times 5)(x^{2+1})(y^{1+3}) = 10 x^3 y^4$ [1]. $x$ and $y$ are different variables; the product rule $x^m \times x^n = x^{m+n}$ only applies when the BASE is the same [1]. So the $x$ indices add together but the $y$ indices add separately, giving two separate variable factors in the answer [1].

Stretch Challenge · +25 XP, +10 coins

Three-Term Expansion

Expand and simplify $-3xy(2x^2 - 4xy + 5y^2)$. Manage every sign carefully; do not combine the three resulting terms.

Reveal solution

$-3xy \times 2x^2 = -6 x^3 y$; $-3xy \times (-4xy) = 12 x^2 y^2$; $-3xy \times 5y^2 = -15 x y^3$. Final: $-6 x^3 y + 12 x^2 y^2 - 15 x y^3$.

Quick Review

Three columns

Coeff, each variable, sign

Two variables

$2x^2y^3 \times 3x^4y = 6x^6 y^4$

Signs

$-2 \times -3 = +6$

Implicit $y^1$

$y$ contributes index $1$

Distribute

$2x(3x^2 - 4x) = 6x^3 - 8x^2$

Check like terms

Different indices stay apart

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