Powers of Ten Notation
Interpret and compare positive numbers in standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer; convert between ordinary notation and standard form
What a learner can do afterwards
- Convert large and small numbers into standard form A × 10ⁿ
- Convert numbers from standard form back to ordinary notation
- Compare and order numbers given in standard form
1 · Read
You have read large numbers in full and used squares and cubes. Standard form writes a positive number as A × 10ⁿ, with 1 ≤ A < 10. This means A is at least 1 and less than 10. The exponent n is an integer, a number with no fractional part. It may be positive, zero or negative. Zero cannot be written in this form. We also write powers using ^: 10^3 means 10³.
To write 7,600,000 in standard form, choose 7.6 for A. Moving its decimal point six places right gives 7,600,000, so 7,600,000 = 7.6 × 10⁶. To convert back, multiply 7.6 by 10 six times. In general, multiplying by 10ⁿ for a positive integer n moves the decimal point n places right. Add zeros where needed.
A negative power of ten means division: 10⁻⁴ = 1 ÷ 10⁴. Multiplying by 10⁻⁴ means dividing by 10 four times, moving the decimal point four places left. Thus 4.5 × 10⁻⁴ = 0.00045. To write 0.00045 in standard form, move its point four places right to make 4.5, then use exponent -4. With 1 ≤ A < 10, a negative exponent gives a positive value below 1.
To compare positive numbers already in standard form, compare the exponents first. The larger exponent gives the larger value because both A values are at least 1 and less than 10. If the exponents match, compare A. To order a mixture of ordinary numbers and standard-form numbers, convert them all to ordinary notation, then compare their place values.
Lowering a power of ten by 1 divides its value by 10: 10² = 100, 10¹ = 10, and 10⁰ = 1. A zero exponent does not give zero. Any nonzero number raised to the power 0 equals 1. So A × 10⁰ = A × 1 = A.
An expression can have the right value without being in standard form. If A is 10 or more, divide A by 10 and add 1 to the exponent. If 0 < A < 1, multiply A by 10 and subtract 1 from the exponent. Repeat until 1 ≤ A < 10. For example, 27 × 10⁴ = 2.7 × 10⁵. Making one factor ten times smaller and the other ten times larger keeps their product unchanged.
To find how much larger one value is, convert both to ordinary numbers and subtract. For example, 6.5 × 10⁴ - 2.5 × 10⁴ = 65,000 - 25,000 = 40,000.
To find how many times as large one positive value is as another, convert both to ordinary numbers and divide. For example, 8 × 10⁵ = 800,000 and 4 × 10⁴ = 40,000. Then 800,000 ÷ 40,000 = 20. The larger value is 20 times as large.
Ratios of small values use the same method: convert, then divide. For example, 9.6 × 10⁻³ = 0.0096 and 8 × 10⁻⁴ = 0.0008. To calculate 0.0096 ÷ 0.0008, multiply both numbers by 10,000. This keeps the quotient unchanged and gives 96 ÷ 8 = 12. The larger value is 12 times as large.
For positive numbers, standard form is A × 10ⁿ with 1 ≤ A < 10 and integer n. Positive powers scale up, negative powers scale down, and 10⁰ = 1.
Questions, readings and flashcards written for this dataset. Linked video: Khan Academy, "Scientific notation examples".
More to read
Checking a standard-form conversion
Standard form separates a positive number into two factors: a coefficient A with 1 ≤ A < 10, and an integer power of ten.
For 48,000, choose A = 4.8. Four multiplications by 10 restore the original value, so 48,000 = 4.8 × 10⁴.
For 0.00063, choose A = 6.3. Four divisions by 10 restore the original value, so 0.00063 = 6.3 × 10⁻⁴. The negative exponent tells you to divide by 10,000. It does not make the value negative.
Check both parts of a conversion: the coefficient must fit the range, and the product must equal the original number.
When comparing exponents works
The exponent-first rule works for positive numbers in standard form because their coefficients are all at least 1 and less than 10.
For example, 1.4 × 10⁵ = 140,000 is larger than 9.6 × 10⁴ = 96,000. The exponent 5 is larger than 4, even though the coefficient 1.4 is smaller than 9.6.
When exponents match, compare coefficients. Thus 2.3 × 10⁻³ is smaller than 7.8 × 10⁻³. Their ordinary values are 0.0023 and 0.0078.
Check the coefficient range before using this rule. For example, 90 × 10² = 9,000 is larger than 1 × 10³ = 1,000 even though 2 is smaller than 3. The expression 90 × 10² is not in standard form.
Equal values and size comparisons
Two expressions can have the same value even when only one is in standard form. For example, 56 × 10³ and 5.6 × 10⁴ both equal 56,000. Dividing the coefficient by 10 and multiplying the power-of-ten factor by 10 leaves the product unchanged.
A difference and a ratio answer different questions. To find how much larger one value is, subtract the smaller from the larger. To find how many times as large it is, divide the larger by the smaller.
For example, 3.5 × 10⁶ = 3,500,000 and 7 × 10⁴ = 70,000. Their difference is 3,430,000. Their ratio is 3,500,000 ÷ 70,000 = 50, so the larger value is 50 times as large.
Explore more
2 · Watch
Take it off screen
Where it sits
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.