Exponents & roots
Exponent Rules
The exponent rules are seven shortcuts for working with powers. When you multiply powers of the same base you add the exponents, when you divide you subtract them, and when you raise a power to a power you multiply them. Any nonzero base to the power 0 is 1, and a negative exponent means a reciprocal. This guide explains each law, why it works, how negative and fractional exponents fit in, and the mistakes that trip people up.
What an exponent means
An exponent says how many times to multiply a base by itself. In 2⁵, the base is 2 and the exponent is 5:
Every exponent rule comes from writing powers out as repeated multiplication like this and counting the factors. If you ever forget a rule, expand a small example and count. That is how each law below is justified, and it is the fastest way to check yourself in an exam.
Exponents come second in the order of operations, after parentheses and before multiplication and division. The PEMDAS calculator shows where powers fit into a longer expression step by step.
To check a small power by hand, a 4 function calculator is enough: 3 × 3 × 3 × 3 gives 81, which is 3⁴.
The seven laws of exponents
In every rule below, a and b are nonzero bases and m and n are exponents.
1. Product rule: add the exponents
Three factors of 2 followed by four more make seven factors in total. The base stays as 2; only the exponents combine.
2. Quotient rule: subtract the exponents
Four factors on the bottom cancel four on the top, leaving two.
3. Power of a power: multiply the exponents
Raising to the power 3 means three copies of 3² multiplied together, and the product rule adds their exponents: three lots of 2 is 6.
4. Power of a product: share the exponent
5. Power of a quotient: share it over the fraction
This is the same as multiplying the fraction by itself four times, which the guide to multiplying fractions covers.
6. Zero exponent: the answer is 1
7. Negative exponent: take the reciprocal
Rules 6 and 7 surprise people the most, so the next two sections explain where they come from.
Why anything to the power of zero is 1
The zero exponent rule is not an arbitrary definition. It is the only value that keeps the other rules working. Two ways to see it:
From the quotient rule
Any number divided by itself is 1, and the quotient rule says the same division gives a⁰. So a⁰ has to be 1.
From the pattern
Each step down in the exponent divides by the base:
Keep dividing by 2 and the pattern runs straight through zero into the negative exponents. Nothing special happens at zero; it is just one more step.
Negative exponents
A negative exponent does not make the answer negative. It means “one over” the positive power. The base flips to the other side of the fraction bar and the exponent loses its minus sign.
| Expression | Rewrite | Value |
|---|---|---|
| 2⁻³ | 1/2³ | 1/8 = 0.125 |
| 5⁻² | 1/5² | 1/25 = 0.04 |
| 3⁻⁴ | 1/3⁴ | 1/81 |
| 10⁻³ | 1/10³ | 1/1,000 = 0.001 |
| (1/2)⁻³ | 2³ | 8 |
| (2/3)⁻² | (3/2)² | 9/4 = 2.25 |
| 1/4⁻² | 4² | 16 |
The last three rows show the move working in reverse. A negative exponent on a fraction flips the fraction. A negative exponent in the denominator moves the power up to the numerator and becomes positive: 1/4⁻² = 4² = 16.
Example: simplify x³ × x⁻⁵, then write it with a positive exponent
All seven rules work with negative exponents exactly as with positive ones. Just be careful with the signs when adding and subtracting exponents; the negative number rules cover 3 + (−5) and −2 − (−6).
Example: simplify (2a⁻²b³)⁻² and write it with positive exponents
Negative powers of ten are the backbone of scientific notation, where 5.2 × 10⁻⁵ means 5.2 divided by 10⁵, or 0.000052.
Fractional exponents and roots
A fractional exponent is a root. The denominator says which root to take and the numerator says which power to raise it to.
Why does a power of 1/2 mean a square root? Apply the power-of-a-power rule: (4^(1/2))² = 4^(1/2 × 2) = 4¹ = 4. So 4^(1/2) is the number that squares to 4, which is exactly what √4 means. The square root calculator handles roots that do not come out whole, such as 2^(1/2), the subject of the square root of 2 guide.
Negative bases and parentheses
A negative base raised to an even power is positive; raised to an odd power it stays negative.
Parentheses decide whether the minus sign is part of the base. Exponents are applied before the negation, so without parentheses only the 3 is squared:
−3² as −9, following the order of operations. If you mean the square of negative three, type the parentheses. The order of operations calculator shows which reading an expression gets.Exponent rules summary table
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1,000 |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)⁴ = 16/81 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | a^(m/n) = (ⁿ√a)ᵐ | 27^(2/3) = 9 |
Exponents also make prime factorization compact: 72 is 2³ × 3² rather than 2 × 2 × 2 × 3 × 3. And the perfect squares chart is a table of n² for every n up to 100.
Common exponent mistakes
- Multiplying the bases.
2³ × 2⁴is2⁷, not4⁷. The base stays the same; only the exponents add. - Multiplying exponents in a product.
x³ × x⁴ = x⁷, notx¹². Exponents multiply only for a power of a power,(x³)⁴ = x¹². - Combining different bases.
2³ × 3²cannot be simplified with the product rule. Work it out:8 × 9 = 72. - Spreading an exponent over addition.
(a + b)² ≠ a² + b². With numbers,(2 + 3)² = 25but2² + 3² = 13. The power-of-a-product rule only works across multiplication. - Treating a negative exponent as a negative number.
2⁻³ = 1/8, which is positive. - Saying a⁰ = 0. Any nonzero base to the power zero is 1.
- Forgetting the parentheses on a negative base.
−3² = −9, while(−3)² = 9.
Frequently asked questions
What are the 7 rules of exponents?
The product rule aᵐ × aⁿ = aᵐ⁺ⁿ, the quotient rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ, the power of a power (aᵐ)ⁿ = aᵐⁿ, the power of a product (ab)ⁿ = aⁿbⁿ, the power of a quotient (a/b)ⁿ = aⁿ/bⁿ, the zero exponent a⁰ = 1, and the negative exponent a⁻ⁿ = 1/aⁿ.
How do negative exponents work?
A negative exponent means take the reciprocal of the positive power: a⁻ⁿ = 1/aⁿ. So 5⁻² = 1/5² = 1/25. The result is not negative; it is a small positive fraction. A negative exponent in a denominator moves up to the numerator, so 1/4⁻² = 4² = 16.
What is 2 to the power of −3?
2⁻³ = 1/2³ = 1/8, which is 0.125 as a decimal. The negative exponent tells you to divide 1 by 2 three times rather than multiply 2 by itself three times. You can see it in the pattern 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8.
Why is any number to the power of 0 equal to 1?
Because it is the only value consistent with the quotient rule. 2⁵ ÷ 2⁵ is 1, since any nonzero number divided by itself is 1, and the quotient rule says it also equals 2⁵⁻⁵ = 2⁰. So 2⁰ = 1. The same argument works for every nonzero base.
Do you add or multiply exponents?
Add them when you multiply two powers with the same base: x³ × x⁴ = x⁷. Multiply them when you raise a power to another power: (x³)⁴ = x¹². Subtract them when you divide powers with the same base: x⁷ ÷ x³ = x⁴. If the bases are different, none of these shortcuts apply.
What does a fractional exponent mean?
A fractional exponent is a root. The denominator gives the root and the numerator gives the power, so a^(m/n) is the nth root of a, raised to the mth power. For example, 27^(2/3) is the cube root of 27, which is 3, squared, which is 9. A power of 1/2 is a square root.
Is −3 squared 9 or −9?
It depends on the parentheses. (−3)² = 9, because negative three is multiplied by itself. −3² = −9, because the exponent is applied to 3 first and the minus sign afterward, following the order of operations. If you mean the square of negative three, write the parentheses.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/exponent-rules/