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How to Use This Calculator
Pick the calculation type from the dropdown. Basic Power calculates xn like 210 = 1,024. Square and Cube are shortcuts for x2 and x3. Square Root and Cube Root calculate √x and ∛x, which equal x1/2 and x1/3. Negative Exponent calculates x-n = 1/(xn), so 2-3 = 1/8. Fractional Exponent takes base, numerator, and denominator to show both methods for xm/n. Scientific Notation converts between a × 10n and standard form. Growth and Decay model exponential change for compound interest, depreciation, or population.
What Are Exponents?
An exponent shows repeated multiplication. In 23, you multiply 2 × 2 × 2 = 8. Read this as "two to the third power" or "two cubed." Just like 3 × 4 means adding 3 four times, 34 means multiplying 3 four times.
In xn, x is the base and n is the exponent. Both can be positive, negative, decimal, or fractional.
Reading Exponents
We say "two squared" for 22 (from the area of a square) and "two cubed" for 23 (from the volume of a cube). Beyond that, we say "to the fourth power," "to the fifth power," and so on. Don't drop the word "power"—saying "two to the fifth" alone is incomplete.
Why Scientists Use Exponents
Exponents compress large and small numbers. The speed of light (300,000,000 m/s) becomes 3 × 108 m/s. An atom's diameter (0.0000000001 m) becomes 1 × 10-10 m. Computer memory uses powers of 2: one kilobyte equals 210 = 1,024 bytes.
Laws of Exponents
xa × xb = x(a+b)
When multiplying the same base, add the exponents. Example: 23 × 24 = 27 = 128. You're multiplying three 2s by four 2s, giving you seven 2s total.
Also: 52 × 53 = 55 = 3,125.
xa ÷ xb = x(a-b)
When dividing the same base, subtract the exponents. 25 ÷ 23 = 22 = 4, which equals 32 ÷ 8.
(xa)b = x(a×b)
When raising a power to a power, multiply the exponents. (23)2 = 26 = 64, which equals 82.
(xy)n = xn × yn
(2×3)2 = 22 × 32 = 4 × 9 = 36, which equals 62.
(x/y)n = xn / yn
(4/2)3 = 43 / 23 = 64 / 8 = 8, which equals 23.
Special Exponent Rules
Zero Exponent
Any nonzero number to the power of zero equals one: x0 = 1. So 50 = 1, 1000 = 1, and (-7)0 = 1.
Using the quotient rule, x3 ÷ x3 = x0. But any number divided by itself equals 1, so x0 must equal 1. The exception is 00, which mathematicians consider undefined.
Negative Exponents
A negative exponent means take the reciprocal: x-n = 1/(xn).
- 2-3 = 1/8 = 0.125
- 10-2 = 1/100 = 0.01
- 5-1 = 1/5 = 0.2
The quotient rule explains this. When you calculate x2 ÷ x5, you get x-3. But x2 ÷ x5 also equals 1/x3, so x-3 = 1/x3.
For fractions, the base flips: (a/b)-n = (b/a)n. So (2/3)-2 = (3/2)2 = 9/4.
Fractional Exponents
A fractional exponent represents a root: x1/n = n√x.
- 81/3 = ∛8 = 2
- 161/4 = 4√16 = 2
- 91/2 = √9 = 3
For xm/n, you can either take the nth root first and then raise to the mth power, or vice versa. Both give the same result. Example: 82/3 = (∛8)2 = 22 = 4, and also ∛(82) = ∛64 = 4.
Negative Base
Even exponents produce positive results: (-2)2 = 4 and (-2)4 = 16. Odd exponents keep the negative sign: (-2)3 = -8 and (-2)5 = -32.
The expression (-2)2 equals 4 because you're squaring -2. But -22 equals -4 because you square 2 first, then apply the negative sign.
Special Bases
One to any power equals one: 15 = 1, 1100 = 1, 1-3 = 1. Zero to any positive power equals zero: 05 = 0, 0100 = 0. However, 00 and 0-n are undefined.
Common Exponent Patterns
Powers of 2
- 24 = 16
- 28 = 256 (one byte in computing)
- 210 = 1,024 (approximately 1,000, used for kilobytes)
- 216 = 65,536
- 232 = 4,294,967,296
Powers of 10
- 103 = 1,000 (thousand)
- 106 = 1,000,000 (million)
- 109 = 1,000,000,000 (billion)
- 1012 = 1,000,000,000,000 (trillion)
Perfect Squares and Cubes
- Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
- Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000
Scientific Notation and Exponents
Scientific notation writes very large or very small numbers using powers of 10. The format is a × 10n, where 1 ≤ a < 10 and n is an integer. So 3,500,000 becomes 3.5 × 106 and 0.000045 becomes 4.5 × 10-5.
Converting to Scientific Notation
For large numbers, move the decimal left and count the places. The number 45,000 becomes 4.5 × 104 because you moved four places. For 7,320,000,000, you move nine places to get 7.32 × 109.
For small numbers, move the decimal right. The number 0.0067 becomes 6.7 × 10-3 because you moved three places. For 0.00000012, you move seven places to get 1.2 × 10-7.
Operations in Scientific Notation
To multiply, multiply the coefficients and add the exponents: (3 × 104) × (2 × 105) = 6 × 109.
To divide, divide the coefficients and subtract the exponents: (8 × 106) ÷ (2 × 103) = 4 × 103.
For addition and subtraction, the exponents must match first. Then add or subtract the coefficients: 3.5 × 104 + 2.1 × 104 = 5.6 × 104.
Real-World Examples
- Speed of light: 3 × 108 m/s
- Earth-Sun distance: 1.5 × 1011 m
- Avogadro's number: 6.02 × 1023
- Electron mass: 9.1 × 10-31 kg
- Planck length: 1.6 × 10-35 m
Exponential Growth and Decay
Growth Formula
The formula A = P(1 + r)n models exponential growth, where A is the final amount, P is the starting amount, r is the growth rate (as a decimal), and n is the number of time periods. This applies to population, investments, and any process that grows by a constant percentage.
A city with 50,000 people growing at 3% annually will have 50,000(1.03)10 = 67,196 people after 10 years. Even though the annual growth is only 3%, the compounding effect produces 34.4% growth over the decade.
An investment of $1,000 at 7% annual return grows to 1,000(1.07)20 = $3,870 after 20 years, nearly quadrupling the initial amount.
Decay Formula
The formula A = P(1 - r)n models exponential decay. Depreciation, radioactive decay, and population decline follow this formula.
A car worth $30,000 that depreciates 15% annually will be worth 30,000(0.85)5 = $13,318 after five years, losing 56% of its value.
Doubling Time and Half-Life
The Rule of 72 estimates doubling time: divide 72 by the growth rate percentage. At 6% growth, doubling time is 72/6 = 12 years. At 9% growth, it's 8 years.
Half-life works the same way for decay, though scientists usually provide this value directly (like Carbon-14's half-life of 5,730 years).
Real-World Uses of Exponents
Compound Interest
The formula A = P(1 + r/n)(nt) calculates compound interest, where P is principal, r is annual rate, n is compounding frequency per year, and t is time in years. Investing $1,000 at 6% compounded monthly for 10 years gives 1,000(1.005)120 = $1,820. Credit card debt uses the same formula—18% APR compounded daily adds up quickly.
Computer Memory
Computers use base 2. One byte equals 28 = 256 values. One kilobyte equals 210 = 1,024 bytes. One megabyte equals 220 = 1,048,576 bytes. One gigabyte equals 230 = 1,073,741,824 bytes. All computer memory capacities are powers of 2.
Radioactive Decay
Carbon-14 dating uses a half-life of 5,730 years. Measure the remaining C-14 and calculate how many half-lives passed. If 25% remains, then (0.5)n = 0.25, so n = 2 half-lives = 11,460 years.
Exponent Tips and Common Mistakes
Calculation Tips
- Break large exponents into smaller ones: 210 = 25 × 25 = 32 × 32 = 1,024
- Use the power rule: (23)3 = 29 = 512
- Powers of 10 are easy: 106 = 1,000,000 (six zeros)
- Each power of 2 doubles the previous: 25 = 32, 26 = 64, 27 = 128
Common Mistakes
-22 = -4 (you square 2 first, then apply the negative)
(-2)2 = 4 (you square the entire -2)
When multiplying same bases add exponents (xa × xb = xa+b), when dividing subtract them (xa / xb = xa-b), and when raising a power to a power multiply them ((xa)b = xab).