1
The rules, in one table.
The opening problem uses the last three, once each. The first two are just as common on the exam.
| When you |
The rule |
Because |
| multiply like bases | am × an = am+n | add |
|---|
| raise a product to a power | (ab)n = anbn | every factor gets the power |
|---|
| raise a power to a power | (am)n = amn | multiply |
|---|
| divide like bases | am ÷ an = am−n | subtract |
|---|
| see a negative power | a−n = 1 ÷ an | move it across the bar |
|---|
The one everyone gets wrong
x−2 is not negative. It is not −x². A negative
exponent says which side of the fraction bar the letter belongs on.
2
Clear the bracket first.
The outside 2 hits every factor inside — the 3 included.
(3x−2y4)2
32 = 9
the number is squared too
9x−4y8
Forgetting to square the 3 is the most common slip here. It is a factor, not a spectator.
3
Divide: subtract the exponents.
9x−4y89x−3y
9 ÷ 9 = 1
the numbers cancel
x−4 − (−3) = x−1
−4 + 3 = −1
y8 − 1 = y7
a lone y is y1
x−1y7
Watch the double negative
Subtracting −3 adds 3. Writing −4 − 3 = −7 here is the single biggest
source of wrong answers in this topic.
4
Move the negative power across the bar.
A finished answer carries no negative exponents.
underneath with a plus
1x
y7x
y7 was already positive, so it stays where it is.
5
Why a zero power is 1.
The rules force it — nobody chose this.
23 ÷ 23 = 8 ÷ 8 = 1
anything over itself
23 ÷ 23 = 23−3 = 20
the subtract rule
so 20 = 1
the same thing, twice
each step down halves — the pattern never breaks
Which is also why a negative exponent means a reciprocal: keep halving past 1.
6
A negative power on a sum.
The rules above are for products and quotients. A sum must be turned into one fraction before an outside power can act on it. Simplify (2a−1 + b−1)−1.
2a−1 + b−1 = 2a + 1b
rewrite each negative power
= 2b + aab
one fraction over ab
(2b + aab)−1 = aba + 2b
the outside −1 flips the fraction
(a + b)−1 is not a−1 + b−1
With numbers: (2 + 3)−1 = 15, but 2−1 + 3−1 = 12 + 13 = 56.
A power never spreads across + or −, so answers like a2 + b are traps.
7
Variations: a negative base, a fraction to a negative power, a negative answer for x.
Simplify (−a2b)3 ÷ (−ab)2.
(−a2b)3 = −a6b3
odd power: the minus stays
(−ab)2 = a2b2
even power: the minus goes
−a6−2b3−2 = −a4b
subtract the exponents
Different: treat the −1 as one more factor in the bracket; its power decides the sign.
Evaluate (23)−2.
(23)−2 = (32)2
flip the fraction, drop the minus
= 94
square top and bottom
Different: a negative power on a fraction just turns it upside down — the answer is bigger than 1, and positive.
The power does not make it negative
−49 (or −94) is the classic wrong answer. A negative exponent moves things across
the bar; it never changes the sign.
Solve 4x = 132.
(22)x = 2−5
4 = 2², and 132 = 2⁻⁵
2x = −5
same base, same exponent
x = −52
check: 4−5/2 = 1√45 = 132 ✓
Different: a fraction on the right means a negative exponent, and the answer for x need not be a whole number.
8
Unknown in the exponent: match the bases.
Solve 9x − 1 = 27x − 3. Write both sides as powers of the same base; then the exponents must be equal.
(32)x − 1 = (33)x − 3
9 = 3², 27 = 3³
32x − 2 = 33x − 9
power to a power: multiply the whole bracket
2x − 2 = 3x − 9
same base, so same exponent
x = 7
check: 96 = 312 and 274 = 312 ✓
The same base trick works when x cannot be found neatly. If 3x = 5, find 9x + 27x.
9x = (32)x = (3x)2 = 25
swap the order of the powers
9x + 27x = 150
x itself was never needed
Powers added or subtracted: factor out the common power. Simplify (3k + 1 + 3k) ÷ (2 · 3k − 1).
3k + 1 + 3k = 3k(3 + 1) = 4 · 3k
3k + 1 = 3k × 3
2 · 3k − 1 = 2 · 3k ÷ 3
3k − 1 = 3k ÷ 3
4 · 3k × 3 ÷ (2 · 3k) = 6
the 3k cancels
The exponent is a whole bracket
(32)x − 1 is 32x − 2, not 32x − 1 — the 2
multiplies both parts of x − 1. Dropping it shifts the answer by one.
Likewise 3x + 2 = 3x × 9, not 3x + 9.
On the exam
Exponent items are usually a simplify-this with a negative exponent inside a
bracket, or an equation with x in the exponent. For the first, test the
choices with x = 2 and y = 3 as in the check below; for the second, rewrite
everything in the smallest base (2, 3 or 5) before doing anything else.
✓
Check: put numbers in.
Pick x = 2 and y = 3, then run both the question and the answer.
the top
(3 · ¼ · 81)² = 3690.5625
x⁻² = ¼, y⁴ = 81
the bottom
9 · ⅛ · 3 = 3.375
x⁻³ = ⅛
divide
3690.5625 ÷ 3.375 = 1093.5
the original
our answer
2187 ÷ 2 = 1093.5
y⁷ ÷ x — a match
Remember the flow
Brackets · Subtract · Move the negatives
No negative exponent survives to the final line.
