Only identical letter-parts can be added. Exponents have their own four rules.
Simplify
5x² + 3x − 2x² + 7x
x² terms combine with x² terms only: 5x² − 2x² = 3x². The x terms combine separately: 3x + 7x = 10x. So the answer is 3x² + 10x, and it cannot be simplified further. A and C add everything together as though x² and x were the same thing — they are not, any more than metres and square metres are. B subtracts 3x from 7x.
| Word | In 7x² − 4x + 9 |
|---|---|
| Terms | 7x², −4x and 9 — three of them |
| Coefficients | 7 and −4 |
| Constant | 9, the term with no letter |
| Like terms | none here — every term differs |
The sign in front belongs to the term. In 7x² − 4x the second term is −4x, not 4x, and that matters the moment you rearrange.
Add the coefficients; the letter part does not change.
Think of the letters as units. 5 metres + 7 metres is 12 metres; 5 square metres + 7 metres is simply two different measurements sitting side by side.
Evaluate 3x² − 2x + 5 at x = −2
the task
3(−2)² − 2(−2) + 5
brackets around the value
= 3(4) + 4 + 5
(−2)² = 4, and −2 × −2 = +4
= 21
12 + 4 + 5
Writing 3 × −2² gives −12, because only the 2 gets squared. Substituting a negative value always goes inside brackets.
| Law | Rule | Example |
|---|---|---|
| Product | same base, multiply → add | x³ · x⁴ = x⁷ |
| Quotient | same base, divide → subtract | x⁸ ÷ x³ = x⁵ |
| Power | a power of a power → multiply | (x³)⁴ = x¹² |
| Zero | anything (not 0) to the 0 → 1 | x⁰ = 1 |
Why x³ · x⁴ = x⁷: three x's times four x's is seven x's in a row. The rules are shortcuts for counting, not arbitrary.
(2x³)⁴
the power hits everything inside
= 2⁴ · x¹²
including the 2
= 16x¹²
not 2x¹²
(4a³)(3a⁶)
a product of two terms
= (4 × 3)(a³ · a⁶) = 12a⁹
numbers multiply, exponents add
Distribute to every term, then collect like terms. Divide numbers with numbers and letters with letters.
Simplify −2(a − 6) + 3a.
−2a + 12 + 3a
−2 × a and −2 × (−6) = +12
= a + 12
−2a + 3a = a
Writing −2(a − 6) as −2a − 12 multiplies the a by −2 but leaves the 6's sign alone, giving a − 12. A negative times a negative is positive, so the term is +12 and the answer is a + 12.
Simplify (18m⁶n²) ÷ (6m²n²).
18 ÷ 6 = 3
numbers with numbers
m⁶ ÷ m² = m⁴ and n² ÷ n² = n⁰ = 1
subtract the exponents for each letter
= 3m⁴
n has cancelled completely
What if the bigger exponent is underneath? Simplify y² ÷ y⁶.
y² ÷ y⁶ = y⁻⁴
2 − 6 = −4
= 1y⁴
a negative exponent means "one over"
Count it out: two y's on top, six underneath, so four y's are left on the bottom. A negative exponent is a reciprocal, never a negative number.
The same rules; these are the values and signs that make them look new.
Evaluate 4x² − x when x = 12.
4(12)² − 12
substitute inside brackets
= 4(14) − 12 = 1 − 12
(12)² = 14
= 12
squaring a fraction below 1 makes it smaller
Fractions follow exactly the same order: exponent first, then multiply, then subtract.
Simplify (−2a²b)³.
(−2)³ · (a²)³ · b³
cube every factor inside
= −8a⁶b³
an odd power keeps the minus
The −2 is a factor like any other: it is cubed to −8, not left as −2 or turned into −6.
Simplify (5x)⁰ + 5x⁰.
(5x)⁰ = 1
the whole bracket is to the zero
5x⁰ = 5 × 1 = 5
only the x is to the zero
1 + 5 = 6
x has gone entirely
Brackets decide what the exponent touches, exactly as with (−2)² and −2².
The exponent laws apply to multiplying and dividing powers, never to adding them. x³ + x⁴ is already as simple as it gets.
Expect a "simplify" with several like terms or a bracket, an "evaluate at a negative value", and an exponent-law item that tests add versus multiply. The fastest route is to work one letter and one exponent at a time; the 10-second check is to put x = 2 into the question and into your answer and see that both give the same number.
Simplify
(3a²b)³
The outside power applies to every factor inside: 3³ = 27, (a²)³ = a⁶ by multiplying the exponents, and b³. So 27a⁶b³. A forgets to cube the 3, the commonest slip here. D adds the exponents (2 + 3) instead of multiplying them, which is the product rule used in the wrong place.
Hint: three separate jobs inside the bracket — the number, the a and the b. Do each one, then put them back together.
