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Understanding Exponents & Powers
Master the 6 fundamental exponent rules
Exponents are a fundamental concept in mathematics, appearing everywhere from compound interest to computer science. An exponent tells you how many times to multiply a number by itself. This guide covers exponent rules, special cases, and practical applications.
What is an Exponent?
In the expression aⁿ, 'a' is the base and 'n' is the exponent (also called power or index).
2 × 2 × 2 = 8
5 × 5 = 25
10 × 10 × 10 × 10
The 6 Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aⁿ × aᵐ = aⁿ⁺ᵐ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient Rule | aⁿ ÷ aᵐ = aⁿ⁻ᵐ | 2⁶ ÷ 2² = 2⁴ = 16 |
| Power Rule | (aⁿ)ᵐ = aⁿˣᵐ | (2³)² = 2⁶ = 64 |
| Zero Exponent | a⁰ = 1 | 5⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional Exponent | a^(1/n) = ⁿ√a | 8^(1/3) = ∛8 = 2 |
Common Powers of 2 and 10
Powers of 2 (Computing)
- 2¹⁰ = 1,024 (1 KB)
- 2²⁰ = 1,048,576 (1 MB)
- 2³⁰ = 1,073,741,824 (1 GB)
- 2⁴⁰ = 1,099,511,627,776 (1 TB)
Powers of 10 (Metric)
- 10³ = 1,000 (Kilo)
- 10⁶ = 1,000,000 (Mega)
- 10⁹ = 1,000,000,000 (Giga)
- 10¹² = 1,000,000,000,000 (Tera)
Frequently Asked Questions
What is an exponent?
An exponent (also called power or index) indicates how many times a number (the base) is multiplied by itself. In aⁿ, 'a' is the base and 'n' is the exponent. For example, 2³ = 2 × 2 × 2 = 8. The exponent tells us to multiply 2 by itself 3 times.
What are the 6 main exponent rules?
The 6 main exponent rules are: 1) Product Rule: aⁿ × aᵐ = aⁿ⁺ᵐ, 2) Quotient Rule: aⁿ ÷ aᵐ = aⁿ⁻ᵐ, 3) Power Rule: (aⁿ)ᵐ = aⁿˣᵐ, 4) Zero Exponent: a⁰ = 1, 5) Negative Exponent: a⁻ⁿ = 1/aⁿ, 6) Fractional Exponent: a^(1/n) = ⁿ√a.
Why does any number to the power of 0 equal 1?
By the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But aⁿ ÷ aⁿ = 1 (any number divided by itself). Therefore, a⁰ = 1. This works for any non-zero number. Note: 0⁰ is typically considered undefined or 1 depending on context.
How do negative exponents work?
A negative exponent means 'one divided by' the positive power. The rule is: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. Negative exponents essentially 'flip' the base into a fraction.
What is a fractional exponent?
A fractional exponent represents a root. The rule is: a^(1/n) = ⁿ√a (nth root of a). For example, 8^(1/3) = ∛8 = 2. More generally, a^(m/n) = (ⁿ√a)ᵐ. So 8^(2/3) = (∛8)² = 2² = 4.
What is 2 to the power of 10?
2¹⁰ = 1,024. This is significant in computing because it represents 1 kilobyte (KB) in binary. Powers of 2 are fundamental in computer science: 2¹⁰ = 1,024 (KB), 2²⁰ = 1,048,576 (MB), 2³⁰ = 1,073,741,824 (GB).
What is 10 to the power of 6?
10⁶ = 1,000,000 (one million). Powers of 10 are used in the metric system and scientific notation: 10³ = 1,000 (kilo), 10⁶ = 1,000,000 (mega), 10⁹ = 1,000,000,000 (giga), 10¹² = 1,000,000,000,000 (tera).
What is scientific notation?
Scientific notation expresses numbers as a × 10ⁿ, where 1 ≤ a < 10. It's used for very large or small numbers. Examples: 5,000,000 = 5 × 10⁶, 0.00003 = 3 × 10⁻⁵. Our calculator automatically shows scientific notation for large results.
Can you raise a negative number to a power?
Yes, with rules: (-a)ⁿ with even n gives positive result, odd n gives negative. Examples: (-2)² = 4, (-2)³ = -8. However, fractional exponents of negative bases (like (-4)^0.5) give complex numbers, which this calculator doesn't handle.
What happens when you raise 0 to a negative power?
0⁻ⁿ is undefined because it equals 1/0ⁿ = 1/0, which is division by zero. Our calculator shows an error for this case. However, 0 to any positive power equals 0.
How do I calculate powers in batch?
Switch to Batch mode and enter expressions in the format 'base^exponent' (one per line). For example: '2^10' or '3^4'. You can also upload a text file. Results show the value and exponent type for each expression.
What is the Product Rule for exponents?
The Product Rule states: aⁿ × aᵐ = aⁿ⁺ᵐ. When multiplying powers with the same base, add the exponents. Example: 2³ × 2⁴ = 2⁷ = 128. This only works when the bases are the same.
What is the Power Rule for exponents?
The Power Rule states: (aⁿ)ᵐ = aⁿˣᵐ. When raising a power to another power, multiply the exponents. Example: (2³)² = 2⁶ = 64. This is different from aⁿ × aᵐ which uses addition.
What is Euler's number (e)?
e ≈ 2.71828... is Euler's number, the base of natural logarithms. It appears in compound interest, exponential growth/decay, and calculus. e¹ = e, e⁰ = 1. The function eˣ is its own derivative, making it fundamental in mathematics.
How accurate is this calculator?
This calculator uses JavaScript's IEEE 754 double-precision arithmetic (about 15-17 significant digits). Very large results (>10³⁰⁸) will show as infinity. You can adjust display precision from 0 to 12 decimal places.