GED Mathematical Reasoning
Exponents, Roots & Scientific Notation
An exponent is shorthand for repeated multiplication: 2⁴ means 2 × 2 × 2 × 2. A square root asks the backwards question — what number times itself gives this? Scientific notation is a way to write very big or very small numbers without a screen full of zeros.
The words first
- base
- the number being multiplied — the 2 in 2⁴
- exponent / power
- how many times to multiply it — the 4 in 2⁴
- square root
- the number that times itself gives you this: √36 = 6
- scientific notation
- a number between 1 and 10, times a power of ten
The method
WHAT AN EXPONENT ACTUALLY SAYS
2⁴ = 2 × 2 × 2 × 2 = 16 (NOT 2 × 4)
5² = 5 × 5 = 25
THE THREE RULES
multiplying, same base → ADD the exponents: x³ · x² = x⁵
dividing, same base → SUBTRACT: x⁵ / x² = x³
power of a power → MULTIPLY: (x³)² = x⁶
NEGATIVE EXPONENT means one over it
2⁻³ = 1/2³ = 1/8
SCIENTIFIC NOTATION
4,200,000 = 4.2 × 10⁶ (dot moved 6 places LEFT → positive 6)
0.00031 = 3.1 × 10⁻⁴ (dot moved 4 places RIGHT → negative 4)
Worked all the way through
The problem
Write 0.00052 in scientific notation.
- Step 1 — Find where the dot has to end up.It must sit right after the first non-zero digit, which is the 5. So the number becomes 5.2.
- Step 2 — Count how many places the dot moved.From 0.00052 to 5.2 the dot moved right: 0.0005.2 — that is 4 places.
- Step 3 — Pick the sign.Moving RIGHT means the original was small, so the exponent is negative: 10⁻⁴.
- Step 4 — Put it together.5.2 × 10⁻⁴
On the board
0.00052
the problem
5.2
dot moved behind the first real digit
4 places, moved right
count the jumps
5.2 × 10⁻⁴
right means a negative exponent
Answer
5.2 × 10⁻⁴
A second one, in a different form
What is 3.4 × 10⁵ in ordinary form?
3.4 × 10⁵
the problem
3.40000
positive 5 → dot moves 5 RIGHT
340,000
Answer
340,000
The usual mistake
Multiplying the bases too. 2³ × 2⁴ is 2⁷, not 4⁷.
Check yourself
Check by moving it back: 5.2 × 10⁻⁴ means move the dot 4 places left → 0.00052. Matches.
Practice it
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