Simplify: \(x^3 \cdot x^4\) → Same base \(x\), so ADD exponents → \(x^{3+4} = x^7\) ✅
1
⭐ Easy
Simplify: \(x^2 \cdot x^5\)
📖 Explanation
Product Rule: Same base → add exponents.
\(x^2 \cdot x^5 = x^{2+5} = x^7\) Trap: Don't multiply the exponents (that's the Power of a Power rule)!
2
⭐ Easy
Evaluate: \(5^0\)
📖 Explanation
Zero Exponent Rule: Any nonzero number to the power of 0 equals 1.
\(5^0 = 1\) Trap: Many students say \(5^0 = 0\) — remember: it's ALWAYS 1!
3
⭐ Easy
Simplify: \(\dfrac{y^8}{y^3}\)
📖 Explanation
Quotient Rule: Same base → subtract exponents (top minus bottom).
\(\dfrac{y^8}{y^3} = y^{8-3} = y^5\) Trap: Don't add — it's subtraction for division!
4
⭐ Easy
Simplify: \((a^3)^4\)
📖 Explanation
Power of a Power Rule: Multiply the exponents.
\((a^3)^4 = a^{3 \times 4} = a^{12}\) Trap: Don't add \(3+4=7\) — this is power of a power, so you MULTIPLY!
5
⭐ Easy
Write with a positive exponent: \(x^{-3}\)
📖 Explanation
Negative Exponent Rule: Flip the base to the denominator.
\(x^{-3} = \dfrac{1}{x^3}\) Trap: Negative exponent does NOT make the number negative!
🔥 Level 2 — Combining Rules
Problems 6–12
✏️ WORKED EXAMPLE — Power of a Product
Simplify: \((2x^3)^2\) → Distribute the power to EACH factor → \(2^2 \cdot (x^3)^2 = 4 \cdot x^6 = 4x^6\) ✅
6
⭐⭐ Medium
Simplify: \((3x^2)^3\)
📖 Explanation
Power of a Product: Apply the exponent to both the coefficient and the variable.
\((3x^2)^3 = 3^3 \cdot (x^2)^3 = 27 \cdot x^{6} = 27x^6\) Trap: Only cubing the variable (getting \(3x^6\)) is the most common mistake!
The parentheses mean the negative sign IS included in the base.
\((-2)^4 = (-2)(-2)(-2)(-2) = 16\) (positive!) Rule: Negative base raised to an EVEN power → POSITIVE result. vs. \(-2^4 = -(2^4) = -16\) — no parentheses, only 2 is the base!
9
⭐⭐ Medium
⚠️ Common Trap
Simplify: \(\dfrac{a^5 b^3}{a^2 b^5}\)
📖 Explanation
Handle each variable separately:
\(a: a^{5-2} = a^3\)
\(b: b^{3-5} = b^{-2} = \dfrac{1}{b^2}\)
Result: \(\dfrac{a^3}{b^2}\) ✅ Trap: \(b^{-2}\) must be written as \(\frac{1}{b^2}\) for positive exponent form.
10
⭐⭐ Medium
Simplify: \(2^3 \cdot 2^{-5}\)
📖 Explanation
Product rule works with negative exponents too — just add!
\(2^3 \cdot 2^{-5} = 2^{3+(-5)} = 2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}\) ✅
11
⭐⭐ Medium
Simplify: \(\left(\dfrac{2}{3}\right)^{-2}\)
📖 Explanation
Negative exponent on a fraction → flip the fraction then apply the positive exponent.
\(\left(\dfrac{2}{3}\right)^{-2} = \left(\dfrac{3}{2}\right)^{2} = \dfrac{9}{4}\) ✅ Key: Negative exponent = reciprocal (flipped fraction)
\((-3)^4 = (-3)(-3)(-3)(-3) = +81\) ← parentheses include the sign
\(-3^4 = -(3^4) = -81\) ← only 3 is the base; negative is applied after
So \(81 > -81\), meaning \((-3)^4\) is larger! ✅
\(a^{m/n} = (\sqrt[n]{a})^m\) — root first, then power.
\(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\) ✅ Memory: denominator = Root, numerator = Power → "R before P"
16
⭐⭐⭐ Tricky
⚠️ Super Common Trap!
Simplify: \(x^2 + x^2\)
📖 Explanation
This is ADDITION, not multiplication!
\(x^2 + x^2 = 2x^2\) (just like \(y + y = 2y\))
Exponent rules (product/quotient) only apply to multiplication and division. ⚠️ The #1 most common student error: treating \(x^2 + x^2\) as \(x^4\)!
Rule used: negative outer exponent flips and squares everything!
📌 MASTER MEMORY CARD
⚡ 8 Rules in 8 Words
1. SAME BASE × → ADD exponents | 2. SAME BASE ÷ → SUBTRACT 3. POWER² → MULTIPLY | 4. ZERO POWER → ONE 5. NEGATIVE POWER → FLIP | 6. PRODUCT POWER → DISTRIBUTE 7. FRACTION POWER → DENOMINATOR=ROOT | 8. ADD ≠ MULTIPLY EXPONENTS