Exponents and Roots

Scientific Notation

Convert between standard form and scientific notation m × 10^n, where 1 ≤ m < 10.

Example

Move the decimal point and record the matching power of ten.

highlighted = computed this step

Step 1 — Large number

Set up the expression.

34003400
Start with 3400; its decimal point is at the end.310^3410^2010^1010^0startMoving left 3 places records a positive power: 10^3.3400 = 3.4 x 10^3

Step 2 — Move decimal left

Move the decimal left 3 places: 3400 = 3.4 x 10^3.

3400=3.4×1033400 = \hl{3.4} \times \hlmath{10^{3}}
Move left 3 places to make 3.4.310^3410^2010^1010^0startcoefficient3 placesMoving left 3 places records a positive power: 10^3.3400 = 3.4 x 10^3

Step 3 — Check expanded value

Check by expanding: 3.4 x 10^3 = 3400.

3.4×103=34003.4 \times 10^{3} = \hl{3400}
3.4 x 10^3 shifts back to 3400.310^3410^2010^1010^0startcoefficient3 placesMoving left 3 places records a positive power: 10^3.3400 = 3.4 x 10^3

Step 4 — Small number

Set up the expression.

0.00720.0072
Start with 0.0072; the decimal point is near the front.010^0010^-1010^-2710^-3210^-4startMoving right 3 places records a negative power: 10^-3.0.0072 = 7.2 x 10^-3

Step 5 — Move decimal right

Move right 3 places: 0.0072 = 7.2 x 10^-3.

0.0072=7.2×10-30.0072 = \hl{7.2} \times 10 ^{ \hl{-3} }
Move right 3 places to make 7.2.010^0010^-1010^-2710^-3210^-4startcoefficient3 placesMoving right 3 places records a negative power: 10^-3.0.0072 = 7.2 x 10^-3

Step 6 — Check expanded value

Check by expanding: 7.2 x 10^-3 = 0.0072.

7.2×10−3=0.00727.2 \times 10 ^{ -3 }= \hl{0.0072}
7.2 x 10^-3 shifts back to 0.0072.010^0010^-1010^-2710^-3210^-4startcoefficient3 placesMoving right 3 places records a negative power: 10^-3.0.0072 = 7.2 x 10^-3

Step 7 — Result

Read the final result.

3.4×1037.2×10−33.4 \times 10^{3} \quad 7.2 \times 10 ^{ -3 }
The exponent records the decimal shift.3400left 3 places3.4x10^30.0072right 3 places7.2x10^-3No extra decimal value is needed; the text is the pinned lesson value.
scientific-notation To write a number in scientific notation: - Move the decimal so 1 ≤ m < 10. - Count moves: right → positive exp, left → negative exp. - Verify: m × 10^n equals the original.