Exponent Calculator

About Exponent Calculator

Calculate any base raised to any exponent and see the result with step-by-step expanded multiplication. Supports positive, negative, zero, and fractional exponents. A built-in properties reference card covers all the rules you need for algebra and beyond.

What Does "Raising to a Power" Mean?

An exponent tells you how many times to multiply the base by itself:

x^n = x × x × x × ... (n times)

Examples:

• 3^4 = 3 × 3 × 3 × 3 = 81
• 2^10 = 1,024
• 5^3 = 125
• 10^6 = 1,000,000

For small integer exponents, the calculator expands every multiplication step. For larger exponents, it shows the formula substitution and final result.

The Eight Exponent Rules

These rules are the foundation of algebra and appear on virtually every standardised maths exam:

Rule	Formula	Example
Zero exponent	x^0 = 1 (x ≠ 0)	7^0 = 1
Identity	x^1 = x	42^1 = 42
Negative exponent	x^(-n) = 1 / x^n	2^(-3) = 1/8 = 0.125
Product of powers	x^a × x^b = x^(a+b)	2^3 × 2^4 = 2^7 = 128
Quotient of powers	x^a / x^b = x^(a-b)	5^6 / 5^2 = 5^4 = 625
Power of a power	(x^a)^b = x^(a×b)	(3^2)^3 = 3^6 = 729
Power of a product	(xy)^n = x^n × y^n	(2×5)^3 = 2^3 × 5^3 = 1,000
Fractional exponent	x^(a/b) = b-th root of x^a	8^(2/3) = (cube root of 8)^2 = 4

How Do Negative Exponents Work?

A negative exponent flips the base to its reciprocal, then applies the positive power:

x^(-n) = 1 / x^n

Worked example: Calculate 4^(-3)

1. Rewrite as a reciprocal: 4^(-3) = 1 / 4^3
2. Calculate 4^3: 4 × 4 × 4 = 64
3. Result: 1/64 = 0.015625

More examples:

Expression	Reciprocal Form	Result
10^(-1)	1/10	0.1
2^(-4)	1/16	0.0625
5^(-2)	1/25	0.04
3^(-3)	1/27	0.037037...

How Do Fractional Exponents Work?

A fractional exponent combines a root and a power. The denominator is the root index, and the numerator is the power:

x^(a/b) = (b-th root of x)^a

Worked example: Calculate 27^(2/3)

1. The denominator 3 means cube root: cube root of 27 = 3
2. The numerator 2 means square: 3^2 = 9
3. Result: 27^(2/3) = 9

Common fractional exponents:

Exponent	Equivalent Root	Example
x^(1/2)	Square root of x	49^(1/2) = 7
x^(1/3)	Cube root of x	64^(1/3) = 4
x^(1/4)	Fourth root of x	81^(1/4) = 3
x^(3/2)	Square root of x, cubed	4^(3/2) = 8
x^(-1/2)	1 / square root of x	9^(-1/2) = 1/3

For dedicated root calculations with radical simplification, the square root calculator shows simplified forms like 6√2.

Powers of Common Bases

These values appear frequently in maths, science, and computing:

n	2^n	3^n	5^n	10^n
0	1	1	1	1
1	2	3	5	10
2	4	9	25	100
3	8	27	125	1,000
4	16	81	625	10,000
5	32	243	3,125	100,000
6	64	729	15,625	1,000,000
7	128	2,187	78,125	10,000,000
8	256	6,561	390,625	100,000,000
9	512	19,683	1,953,125	1,000,000,000
10	1,024	59,049	9,765,625	10,000,000,000

Powers of 2 are especially important in computer science. 2^10 = 1,024 is approximately one thousand (1 KB), 2^20 ≈ one million (1 MB), and 2^30 ≈ one billion (1 GB).

The Special Case of 0^0

Zero raised to the zero power is one of the most debated expressions in maths. In combinatorics and most programming languages, 0^0 is defined as 1. This convention makes formulas like the binomial theorem and power series work cleanly. However, in analysis and calculus, 0^0 is considered an indeterminate form because different limits can approach different values.

The calculator flags 0^0 as a special case and shows the conventional result of 1 with an explanation.

Where Exponents Appear in Real Life

Application	Formula	How Exponents Are Used
Compound interest	A = P(1 + r/n)^(nt)	Money grows exponentially over time
Population growth	P = P₀ × e^(rt)	Bacteria double at a fixed rate
Radioactive decay	N = N₀ × (1/2)^(t/h)	Half-life means halving repeatedly
Computer storage	2^10, 2^20, 2^30	KB, MB, GB are all powers of 2
Sound intensity	dB = 10 × log₁₀(I/I₀)	Decibels use logarithmic (inverse of exponential) scale
Earthquake magnitude	Each +1 = 10× energy	Richter scale is a power of 10 scale

Exponents vs Logarithms

Exponents and logarithms are inverse operations. If you know that 2^8 = 256, then log₂(256) = 8. Exponents answer "what do I get when I multiply the base by itself n times?" while logarithms answer "how many times must I multiply the base to reach this number?"

Exponential Form	Logarithmic Form
2^8 = 256	log₂(256) = 8
10^3 = 1,000	log₁₀(1,000) = 3
e^1 = 2.718	ln(2.718) = 1
5^4 = 625	log₅(625) = 4

For logarithm calculations, the log calculator supports any base with step-by-step working. To express results in scientific notation, the scientific notation calculator handles conversions and arithmetic.

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