📖 What Is an Exponent?
An exponent tells you how many times to multiply a number (or variable) by itself. The number being multiplied is called the base. The small number written above it is the exponent.
For example: x3 means x · x · x — the base is x, multiplied 3 times.
The Base
The number or variable being multiplied
In x4, the base is x.
In 53, the base is 5.
In (½)2, the base is ½.
The Exponent
The small number — how many times to multiply
In x4, the exponent is 4.
In 53, the exponent is 3.
In (½)2, the exponent is 2.
Focus on the base first. On the GED, exponent problems often look scary because the exponents are large. But the rule you need only depends on the base — not the size of the exponents.
✖️ Rule 1 — Multiplying: Add the Exponents
When you multiply two expressions that have the same base, you keep the base and add the exponents.
Why does this work? Think about it: x3 · x4 means (x · x · x) · (x · x · x · x) — that's 7 x's being multiplied, which is x7. You're just counting them up.
Important: The base stays the same. You do NOT multiply the bases. Only the exponents get added.
Common mistake to avoid: x3 · x4 ≠ x12 — don't multiply the exponents when you're multiplying. That's a different rule (powers of powers, below).
➗ Rule 2 — Dividing: Subtract the Exponents
When you divide two expressions with the same base, keep the base and subtract the bottom exponent from the top.
Why does this work? Division cancels out matching factors. x7 ÷ x3 means you cancel 3 of the x's, and 4 are left — so you get x4.
Order matters! Always subtract top minus bottom: x7 ÷ x3 = x7−3 = x4.
Not x3−7 — that would give a negative exponent (not tested on GED).
📦 Rule 3 — Power Raised to a Power
When an exponent expression is inside parentheses and raised to another power, you multiply the exponents.
Example: (x2)3 = x2×3 = x6
This works because (x2)3 means x2 · x2 · x2 — that's 3 groups of 2, or 6 total.
The parentheses are the signal. See (xa)b with parentheses? Multiply. See xa · xb without? Add.
🔑 The Base Can Be Anything
These rules work the same way whether the base is a variable, a whole number, or a fraction. The rules don't change — just look for the same base.
Variable Base
x3 · x5 = x8
Same base (x) — add the exponents.
Number Base
53 · 52 = 55
Same base (5) — add the exponents.
You don't need to calculate 55.
Fraction Base
(12)4 · (12)2 = (12)6
Same base (½) — add the exponents.
The fraction is still just "the base."
Key Insight
Simplify the exponent — not the arithmetic
On the GED, you usually write the simplified exponent expression as your answer. You rarely need to calculate the full number.
GED strategy: These problems look scary because of big exponents like x12 or 315. But you never compute the big number — you just apply the rule and write the simplified expression. That's the whole trick.
🔢 Worked Examples
Example 1 — Multiplying Variables
Simplify: x4 · x6
Step 1: Identify the base → both have base x. Same base? Use the multiply rule.
Step 2: Add the exponents → 4 + 6 = 10
Step 3: Write the answer: x10
Example 2 — Dividing Variables
Simplify: x9 ÷ x3
Step 1: Identify the base → both have base x. Dividing? Subtract the exponents.
Step 2: Subtract top minus bottom → 9 − 3 = 6
Step 3: Write the answer: x6
Example 3 — Multiplying Numbers
Simplify: 23 · 25
Step 1: Base is 2 on both sides. Multiplying → add the exponents.
Step 2: 3 + 5 = 8
Answer: 28 — you don't need to compute 256.
Example 4 — Power Raised to a Power
Simplify: (x3)4
Step 1: See parentheses with an outer exponent → multiply rule.
Step 2: Multiply the exponents → 3 × 4 = 12
Answer: x12
Example 5 — Division with Numbers (the "scary" type)
Simplify: 3835
Step 1: Same base (3). Division → subtract. 8 − 5 = 3
Answer: 33 — or 27 if you need the number.
This problem looks complicated but the rule takes 2 seconds.
✏️ Practice Questions