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GED Math Preparation
🔢 Simplifying Expressions with Exponents
Laws of exponents — power of a power, product of powers & more
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⚡
Rule 1 — Power of a Power
(aᵐ)ⁿ = aᵐˣⁿ — multiply exponents
✖️
Rule 2 — Product of Powers
aᵐ · aⁿ = aᵐ⁺ⁿ — add exponents
🚫
Rule 3 — Same Base Only
Only combine when bases match
📤
Rule 4 — Distribute Exponent
(a·b)ⁿ = aⁿ·bⁿ
🧩
GED Worked Example
Solve (5³·2⁴)²(5⁻²·2⁵) step by step
🔧
Interactive Calculator
Enter your own exponent problems

💡 What You Will Learn

Exponent rules let you simplify complex expressions like (5³·2⁴)²(5⁻²·2⁵) into a clean answer like 5⁴·2¹³. There are just 4 rules to master — and these appear constantly on the GED!

Rule 1 — Power of a Power
When a power already has an exponent — multiply the exponents.
The Formula
(am)n = am × n
Exponent raised to another exponent → MULTIPLY them
1
Identify the inner and outer exponents

(53)2 — inner exponent is 3, outer exponent is 2.

2
Multiply the exponents

3 × 2 = 6 → 56

3
Example: (2⁴)²

4 × 2 = 8 → 28

Example 1
(53)2 = 56
Example 2
(24)2 = 28
Example 3
(x3)4 = x12
Example 4
(a2)5 = a10
🧠 Try It

What is (34)3? (Multiply: 4 × 3 = ?)

Rule 2 — Product of Powers
Multiplying the same base — add the exponents.
The Formula
am · an = am + n
Same base multiplied → ADD the exponents
1
Check the bases match

56 · 5−2 — both base 5. ✅

2
Add the exponents (including negatives!)

56 · 5−2 = 56+(−2) = 54

3
Example: 2⁸ · 2⁵

28+5 = 213

Example 1
56 · 5−2 = 54
Example 2
28 · 25 = 213
Example 3
a5 · a2 = a7
Example 4
x3 · x−1 = x2
🧠 Try It

What is 73 · 74? (Add: 3 + 4 = ?)

Rule 3 — Same Base Only
You can ONLY combine exponents when the bases are identical!
📌 The Rule

5's combine with 5's. 2's combine with 2's. You cannot combine a 5 and a 2 — even if both have exponents. Keep different bases separate!

✅
CAN combine — same base

53 · 54 = 57  |  26 · 22 = 28

🚫
CANNOT combine — different bases

52 · 23 stays as 5² · 2³ — you CANNOT write this as 105!

⚠️ Most Common GED Mistake

Wrong: 52 · 23 = 105 ❌

Right: 52 · 23 = 5² · 2³ ✅ — leave them separate!

✅ CAN Combine
53 · 52 = 55
Same base (5)
🚫 CANNOT Combine
53 · 22 = 5³·2²
Different bases
🧠 Quick Check

Can you simplify 34 · 72 into a single term?

Rule 4 — Distribute the Exponent
An outside exponent applies to EVERY factor inside the parentheses.
The Formula
(a · b)n = an · bn
Outside exponent goes to each factor inside
1
Identify what's inside and the outside exponent

(53 · 24)2 — inside: 5³ and 2⁴, outside: ²

2
Apply the exponent to EACH factor

(53)2 · (24)2 — the ² goes to both!

3
Use Rule 1 (Power of Power) on each

(53)2=56   (24)2=28 → 56 · 28

Example 1
(a·b)3 = a3·b3
Example 2
(5³·2⁴)2 = 56·28
Example 3
(x²·y³)4 = x8·y12
Example 4
(2·3)2 = 4·9 = 36
🧠 Try It

What does (x3 · y2)3 simplify to?

GED Worked Example — Step by Step
The exact problem from the GED screenshot — answer is 5⁴ · 2¹³
Simplify: (5³ · 2⁴)² (5⁻² · 2⁵)
(53 · 24)2 (5−2 · 25)
Step A — Distribute the outside ² (Rule 4)
(53)2 · (24)2 · 5−2 · 25
² goes to each factor inside
Step B — Power of a Power (Rule 1): multiply exponents
(53)2 = 56
3 × 2 = 6
(24)2 = 28
4 × 2 = 8
→
56 · 28 · 5−2 · 25
All four terms written out
Step C — Group same bases (Rule 3)
56 · 5−2
5's together
28 · 25
2's together
Step D — Product of Powers (Rule 2): add exponents
56+(−2) = 54
6 + (−2) = 4
28+5 = 213
8 + 5 = 13
🏆
Final Answer — matches GED answer key!
(5³·2⁴)²(5⁻²·2⁵) = 5⁴ · 2¹³
Rules Used in This Problem
RuleFormulaWhere Used
Distribute Exponent(ab)ⁿ=aⁿbⁿStep A: ² to 5³ and 2⁴
Power of a Power(aᵐ)ⁿ=aᵐˣⁿStep B: (5³)²=5⁶, (2⁴)²=2⁸
Same Base OnlyCombine matching basesStep C: 5's with 5's, 2's with 2's
Product of Powersaᵐ·aⁿ=aᵐ⁺ⁿStep D: 5⁶·5⁻²=5⁴, 2⁸·2⁵=2¹³
Interactive Exponent Calculator
Enter values and see the rule applied instantly!
⚡ Power of a Power — (aᵐ)ⁿ
Base (a):
Inner exp (m):
Outer exp (n):
✖️ Product of Powers — aᵐ · aⁿ
Base (a):
First exp (m):
Second exp (n):
🧠 Challenge

Try the Power of Power tool with base=5, m=3, n=2. Then use Product of Powers with base=5, m=6, n=−2. These are the exact steps from the GED example!

Practice Exercises
Apply the rules — then reveal the solution!
Question 1 of 8
Score: 0 / 8
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