TOPIC 01
Law of Exponents — Multiplication
am × an = am+n
★ Quick Memory Point
SAME BASE → ADD exponents
Think: "Same Base, Sum the Power" (SBSP)
⚠️ Only works when the base is identical. Never add exponents of different bases! am × an = am+n | 23 × 25 = 28
Think: "Same Base, Sum the Power" (SBSP)
⚠️ Only works when the base is identical. Never add exponents of different bases! am × an = am+n | 23 × 25 = 28
Q 01
medium
Simplify: 34 × 36
✓ Correct Answer: A — 310
Apply the Product Rule: same base → add the exponents.
34 × 36 = 34+6 = 310
KEY STEP: 4 + 6 = 10 (add, do NOT multiply)
Apply the Product Rule: same base → add the exponents.
34 × 36 = 34+6 = 310
KEY STEP: 4 + 6 = 10 (add, do NOT multiply)
Q 02
medium
Which is equal to x3 · x5 · x2?
✓ Correct Answer: C — x10
Add ALL exponents: 3 + 5 + 2 = 10.
The coefficient stays 1; do not multiply 3 bases into 3x10. x3 · x5 · x2 = x3+5+2 = x10
Add ALL exponents: 3 + 5 + 2 = 10.
The coefficient stays 1; do not multiply 3 bases into 3x10. x3 · x5 · x2 = x3+5+2 = x10
Q 03
hard
Find the value of n if 2n × 24 = 212.
✓ Correct Answer: B — n = 8
Set up: n + 4 = 12 → n = 12 − 4 = 8.
Common mistake: students pick n = 3 (dividing 12 ÷ 4). Wrong — exponents ADD, not multiply. n + 4 = 12 → n = 8
Set up: n + 4 = 12 → n = 12 − 4 = 8.
Common mistake: students pick n = 3 (dividing 12 ÷ 4). Wrong — exponents ADD, not multiply. n + 4 = 12 → n = 8
Q 04
tricky ★
Simplify: a2 · b3 · a5 · b
⚡ Tricky — two different bases!
✓ Correct Answer: A — a7b4
Group by base: a2 · a5 = a7 and b3 · b1 = b4.
Remember b = b1! The exponent 1 is invisible but real. a: 2+5=7 | b: 3+1=4 → a7b4
Group by base: a2 · a5 = a7 and b3 · b1 = b4.
Remember b = b1! The exponent 1 is invisible but real. a: 2+5=7 | b: 3+1=4 → a7b4
TOPIC 02
Law of Exponents — Division
am ÷ an = am−n
★ Quick Memory Point
SAME BASE, DIVIDE → SUBTRACT exponents
Think: "Divide = Minus" (D=M)
⚠️ Top exponent MINUS bottom exponent. Order matters! am ÷ an = am−n | 57 ÷ 53 = 54
Think: "Divide = Minus" (D=M)
⚠️ Top exponent MINUS bottom exponent. Order matters! am ÷ an = am−n | 57 ÷ 53 = 54
Q 05
medium
Simplify: x9 ÷ x4
✓ Correct Answer: B — x5
Division rule: subtract exponents. 9 − 4 = 5.
Don't add (A) or divide (C) the exponents — subtract! x9 ÷ x4 = x9−4 = x5
Division rule: subtract exponents. 9 − 4 = 5.
Don't add (A) or divide (C) the exponents — subtract! x9 ÷ x4 = x9−4 = x5
Q 06
medium
What is
a8
a3
?
✓ Correct Answer: A — a5
a8 ÷ a3 = a8−3 = a5.
Fraction form = division. Always subtract (numerator − denominator). 8 − 3 = 5 → a5
a8 ÷ a3 = a8−3 = a5.
Fraction form = division. Always subtract (numerator − denominator). 8 − 3 = 5 → a5
Q 07
tricky ★
Simplify:
12x6
4x2
⚡ Don't forget to divide the coefficients too!
✓ Correct Answer: B — 3x4
Divide coefficient: 12 ÷ 4 = 3.
Divide variable: x6 ÷ x2 = x6−2 = x4.
Do BOTH steps — many students forget to divide the number! 12÷4=3 and 6−2=4 → 3x4
Divide coefficient: 12 ÷ 4 = 3.
Divide variable: x6 ÷ x2 = x6−2 = x4.
Do BOTH steps — many students forget to divide the number! 12÷4=3 and 6−2=4 → 3x4
TOPIC 03
Power of a Power
(am)n = am×n
★ Quick Memory Point
POWER of a POWER → MULTIPLY exponents
Think: "Brackets? MULTIPLY inside"
⚠️ This is the #1 confusion point: (am)n is multiply, NOT add! (am)n = am×n | (32)4 = 38
Think: "Brackets? MULTIPLY inside"
⚠️ This is the #1 confusion point: (am)n is multiply, NOT add! (am)n = am×n | (32)4 = 38
Q 08
medium
Simplify: (23)4
✓ Correct Answer: C — 212
Power of power: multiply exponents. 3 × 4 = 12.
✗ A: 3+4=7 (added — wrong rule) | ✗ D: 43=64 (nonsense). (23)4 = 23×4 = 212
Power of power: multiply exponents. 3 × 4 = 12.
✗ A: 3+4=7 (added — wrong rule) | ✗ D: 43=64 (nonsense). (23)4 = 23×4 = 212
Q 09
medium
Simplify: (x5)3
✓ Correct Answer: A — x15
5 × 3 = 15. Multiply, don't add (8) or concatenate (53)! (x5)3 = x5×3 = x15
5 × 3 = 15. Multiply, don't add (8) or concatenate (53)! (x5)3 = x5×3 = x15
Q 10
tricky ★
Which equals (2x3)4?
⚡ The coefficient gets raised too!
✓ Correct Answer: D — 16x12
Apply the power to every factor: 24 = 16 and (x3)4 = x12.
Most common mistake: forgetting to raise the coefficient 2 to the power 4. 24=16, x3×4=x12 → 16x12
Apply the power to every factor: 24 = 16 and (x3)4 = x12.
Most common mistake: forgetting to raise the coefficient 2 to the power 4. 24=16, x3×4=x12 → 16x12
TOPIC 04
Zero & Negative Exponents
a0 = 1 | a−n = 1 / an
★ Quick Memory Point
ZERO exponent → always 1 (as long as base ≠ 0)
NEGATIVE exponent → FLIP to denominator
Think: "Negative = Flip (reciprocal)"
⚠️ (−2)0 = 1 but −20 = −1. Watch the parentheses! 50=1 | 2−3=1/8 | x−2=1/x2
NEGATIVE exponent → FLIP to denominator
Think: "Negative = Flip (reciprocal)"
⚠️ (−2)0 = 1 but −20 = −1. Watch the parentheses! 50=1 | 2−3=1/8 | x−2=1/x2
Q 11
medium
Evaluate: 70 + 30
✓ Correct Answer: B — 2
70 = 1 and 30 = 1, so 1 + 1 = 2.
✗ D is 70 alone. Read carefully — it's a SUM of two terms. Any base0 = 1 → 1 + 1 = 2
70 = 1 and 30 = 1, so 1 + 1 = 2.
✗ D is 70 alone. Read carefully — it's a SUM of two terms. Any base0 = 1 → 1 + 1 = 2
Q 12
medium
Evaluate: 2−3
✓ Correct Answer: C — 1/8
Negative exponent = reciprocal. 2−3 = 1/23 = 1/8.
✗ A: negative base is a different thing entirely. The base (2) stays positive! 2−3 = 1/23 = 1/8
Negative exponent = reciprocal. 2−3 = 1/23 = 1/8.
✗ A: negative base is a different thing entirely. The base (2) stays positive! 2−3 = 1/23 = 1/8
Q 13
tricky ★
Which is correct?
⚡ Classic trap — watch the parentheses!
✓ Correct Answer: B — (−3)0 = 1
(−3)0: the base is −3 → result = 1.
−30: only 3 is raised to 0 → 30=1, then − → equals −1.
Parentheses completely change the meaning! ( )0=1 vs −(30)=−1
(−3)0: the base is −3 → result = 1.
−30: only 3 is raised to 0 → 30=1, then − → equals −1.
Parentheses completely change the meaning! ( )0=1 vs −(30)=−1
Q 14
hard
Simplify: x−4 · x7
✓ Correct Answer: C — x3
Same base, multiply → add exponents: −4 + 7 = 3.
Negative exponent just means the number is negative when adding: −4 + 7 = +3. x−4 · x7 = x−4+7 = x3
Same base, multiply → add exponents: −4 + 7 = 3.
Negative exponent just means the number is negative when adding: −4 + 7 = +3. x−4 · x7 = x−4+7 = x3
TOPIC 05
Mixed & Combined Laws
Apply rules in correct order · most common exam traps
★ Quick Memory Point — The 4 Laws at a Glance
① SAME BASE × → ADD
② SAME BASE ÷ → SUBTRACT
③ POWER of POWER → MULTIPLY ④ ZERO → ONE, NEGATIVE → FLIP
Master these 4 rules and you own exponents. × = + | ÷ = − | ( )n = × | 0=1, −=flip
③ POWER of POWER → MULTIPLY ④ ZERO → ONE, NEGATIVE → FLIP
Master these 4 rules and you own exponents. × = + | ÷ = − | ( )n = × | 0=1, −=flip
Q 15
hard
Simplify: (a3b2)3
✓ Correct Answer: A — a9b6
Distribute the outer exponent 3 to each factor inside:
a3×3 = a9 and b2×3 = b6.
✗ D: The 3 outside is an exponent, not a coefficient multiplier. Distribute outer power to EVERY factor inside
Distribute the outer exponent 3 to each factor inside:
a3×3 = a9 and b2×3 = b6.
✗ D: The 3 outside is an exponent, not a coefficient multiplier. Distribute outer power to EVERY factor inside
Q 16
tricky ★
Simplify:
(x4)3
x5
⚡ Step 1: simplify numerator first!
✓ Correct Answer: B — x7
Step 1: (x4)3 = x12 (power of power → multiply)
Step 2: x12 ÷ x5 = x12−5 = x7 (division → subtract)
Always handle brackets FIRST. ①(x4)3=x12 → ②x12÷x5=x7
Step 1: (x4)3 = x12 (power of power → multiply)
Step 2: x12 ÷ x5 = x12−5 = x7 (division → subtract)
Always handle brackets FIRST. ①(x4)3=x12 → ②x12÷x5=x7
Q 17
hard
Evaluate: 32 × 33 ÷ 34
✓ Correct Answer: A — 3 (= 31)
Work left to right: 32+3 = 35, then 35−4 = 31 = 3.
Or combine all: exponents = 2+3−4 = 1 → 31 = 3. 2+3−4=1 → 31=3
Work left to right: 32+3 = 35, then 35−4 = 31 = 3.
Or combine all: exponents = 2+3−4 = 1 → 31 = 3. 2+3−4=1 → 31=3
Q 18
hard
If 4x = 64, find x.
⚡ Write 64 as a power of 4!
✓ Correct Answer: C — x = 3
64 = 4 × 4 × 4 = 43. So 4x = 43 → x = 3.
Key skill: convert both sides to the same base, then compare exponents. 64=43 → same base → x=3
64 = 4 × 4 × 4 = 43. So 4x = 43 → x = 3.
Key skill: convert both sides to the same base, then compare exponents. 64=43 → same base → x=3
Q 19
tricky ★
Simplify: (3−2)−1
⚡ Negative × Negative = Positive!
✓ Correct Answer: B — 9
(3−2)−1 = 3(−2)×(−1) = 32 = 9.
Negative × negative = positive. The result is NOT negative. (−2)×(−1)=+2 → 32=9
(3−2)−1 = 3(−2)×(−1) = 32 = 9.
Negative × negative = positive. The result is NOT negative. (−2)×(−1)=+2 → 32=9
Q 20
tricky ★
Which expression is equivalent to
(2a2b)3
4a3b2
?
⚡ Boss-level: expand numerator first, then divide!
✓ Correct Answer: A = C — 2a3b
Step 1: (2a2b)3 = 23·a6·b3 = 8a6b3
Step 2: 8a6b3 ÷ 4a3b2
coefficient: 8÷4 = 2
a: 6−3 = a3
b: 3−2 = b1 = b
Result: 2a3b ①Expand →8a6b3 ②Divide each part → 2a3b
Step 1: (2a2b)3 = 23·a6·b3 = 8a6b3
Step 2: 8a6b3 ÷ 4a3b2
coefficient: 8÷4 = 2
a: 6−3 = a3
b: 3−2 = b1 = b
Result: 2a3b ①Expand →8a6b3 ②Divide each part → 2a3b
Quiz Complete 🎉
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