Algebraic Expressions and Index Laws
Apply product, quotient and power rules to algebraic bases. Know the difference between like terms (add) and same-base multiplication (multiply).
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Decide: are $3x^2$ and $4x^2$ like terms? What happens when you compute $3x^2 + 4x^2$ versus $3x^2 \times 4x^2$? Predict both answers before reading on.
The index laws work for any base, including a letter. So $x^3 \times x^4 = x^7$ and $m^6 \div m^2 = m^4$. With coefficients, multiply the numbers and apply the index law to the variables.
$x^m \times x^n = x^{m+n}$. With coefficients, $a x^m \times b x^n = (ab)\, x^{m+n}$. The coefficients multiply, the indices add. Like-term addition $a x^m + b x^m = (a+b) x^m$ keeps the index the same.
Know
- $x^m \times x^n = x^{m+n}$
- $x^m \div x^n = x^{m-n}$
- $(x^m)^n = x^{mn}$
Understand
- Why coefficients are separate from indices
- Why $3x^2 + 4x^2 \ne 7x^4$
- Why the operation symbol decides everything
Can Do
- Simplify $3x^2 \times 4x^3 = 12x^5$
- Add $3x^2 + 4x^2 = 7x^2$
- Decide which operation is needed
Wrong: “$3x^2 + 4x^2 = 7x^4$” — adding the indices when adding like terms.
Right: $3x^2 + 4x^2 = 7x^2$. Index stays the same when adding like terms.
Wrong: “$3x^2 \times 4x^2 = 7x^4$” — adding the coefficients when multiplying.
Right: $3x^2 \times 4x^2 = 12 x^4$. Coefficients $3 \times 4 = 12$, indices $2 + 2 = 4$.
Letters behave exactly like numbers under the index laws — the symbol just stays as a letter.
$x^3 \times x^4 = x^{3+4} = x^7$. $m^6 \div m^2 = m^{6-2} = m^4$. $(p^3)^5 = p^{3 \times 5} = p^{15}$. Indices behave identically whether the base is a number or a letter.
To multiply $3x^2 \times 4x^3$: handle coefficients and variables separately. Multiply the numbers, then apply the product rule to the variables.
$3x^2 \times 4x^3 = (3 \times 4)(x^{2+3}) = 12 x^5$. The number-with-number step uses normal multiplication; the variable-with-variable step uses the product rule.
Watch Me Solve It · 3 examples
- 1Separate coefficients from variables$3x^2 \times 4x^3 = (3 \times 4) \times (x^2 \times x^3)$
- 2Multiply coefficients, add indices$12 \times x^{2+3} = 12 x^5$Product rule for the variables.
- 3Final$12x^5$
- 1(a) Identify like terms$3x^2 + 4x^2$ — same base $x$, same index $2$Add the coefficients only.
- 2(a) Add coefficients$3x^2 + 4x^2 = 7x^2$
- 3(b) Multiply terms$3x^2 \times 4x^2 = 12 x^{2+2} = 12x^4$Coefficients $\times$, indices $+$.
- 1(a) Quotient rule on a letter$m^6 \div m^2 = m^{6-2} = m^4$
- 2(b) Power-of-a-power on a letter$(p^3)^5 = p^{3 \times 5} = p^{15}$
- 3Check$m^4$ and $p^{15}$Letters behave like numbers.
Common Pitfalls
Multiply terms
- $3x^2 \times 4x^3 = 12x^5$
- Coefficients: multiply
- Indices: add
Add like terms
- $3x^2 + 4x^2 = 7x^2$
- Coefficients: add
- Index stays
Pure variable rules
- $x^3 \times x^4 = x^7$
- $m^6 \div m^2 = m^4$
- $(p^3)^5 = p^{15}$
Decision flow
- See $+/-$? Add coefficients
- See $\times$? Multiply & add indices
- Same base for either
How are you completing this lesson?
Brain Trainer · 4 problems
Decide the operation first, then simplify.
1 Simplify $5x^4 \times 2x^3$.
Coefficients $5 \times 2 = 10$; indices $4 + 3 = 7$.$10 x^7$2 Simplify $5x^4 + 2x^4$.
Like terms — add coefficients; index stays $4$.$7 x^4$3 Simplify $(y^4)^2$.
Power of a power: $4 \times 2 = 8$.$y^8$4 Simplify $\dfrac{8 m^7}{2 m^2}$.
Coefficients $8 \div 2 = 4$; indices $7 - 2 = 5$.$4 m^5$
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Simplify each: (a) $x^4 \times x^3$, (b) $(y^2)^6$, (c) $\dfrac{a^9}{a^4}$.
Q7. Simplify, paying attention to the operation: (a) $6x^3 + 4x^3$, (b) $6x^3 \times 4x^2$, (c) $\dfrac{20 m^8}{5 m^3}$.
Q8. A student wrote “$2x^2 + 5x^2 = 7x^4$.” Explain the mistake, give the correct answer, and write one similar example where the index DOES change, including which operation causes that.
Quick Check
1. C — $12x^5$.
2. A — $7x^2$.
3. D — $p^{15}$.
4. B — $m^4$.
5. A — $12x^4$.
Show Your Working Model Answers
Q6 (3 marks): (a) $x^{4+3} = x^7$ [1]; (b) $y^{2 \times 6} = y^{12}$ [1]; (c) $a^{9-4} = a^5$ [1].
Q7 (3 marks): (a) like terms, add coefficients: $6x^3 + 4x^3 = 10x^3$ [1]; (b) coefficients $6 \times 4 = 24$, indices $3 + 2 = 5$: $24x^5$ [1]; (c) coefficients $20 \div 5 = 4$, indices $8 - 3 = 5$: $4m^5$ [1].
Q8 (3 marks): Mistake: the student added indices when adding LIKE TERMS. Like-term addition keeps the index, so $2x^2 + 5x^2 = 7x^2$ [1]. The index only changes when MULTIPLYING (product rule) [1]. Example: $2x^2 \times 5x^2 = 10x^4$ — here the operation $\times$ causes indices to add [1].
Operation Detective
Simplify completely: $2x^3 + 3x^3 + 2x^3 \times 3x^2$. Show each operation clearly; remember that $\times$ binds before $+$.
Reveal solution
Multiplication first: $2x^3 \times 3x^2 = 6x^5$. Then add like terms: $2x^3 + 3x^3 = 5x^3$, so the answer is $5x^3 + 6x^5$. They are NOT like terms (different indices), so they cannot combine further.
Product rule
$x^m \times x^n = x^{m+n}$
Quotient rule
$x^m \div x^n = x^{m-n}$
Power rule
$(x^m)^n = x^{mn}$
Like terms
$3x^2 + 4x^2 = 7x^2$
Coefficients $\times$
$3x^2 \times 4x^3 = 12x^5$
Read the operation
$+/-$ vs $\times$ matters
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