Powers & Roots Calculator
Calculate any power or root — squares, cubes, nth roots, negative exponents and fractional exponents — with step-by-step working.
POWERS & ROOTS CALCULATOR — STEP BY STEP
Choose a mode, enter your values and press Calculate
Reference tables
Perfect squares
| 1² | = | 1 |
| 2² | = | 4 |
| 3² | = | 9 |
| 4² | = | 16 |
| 5² | = | 25 |
| 6² | = | 36 |
| 7² | = | 49 |
| 8² | = | 64 |
| 9² | = | 81 |
| 10² | = | 100 |
| 11² | = | 121 |
| 12² | = | 144 |
Perfect cubes
| 1³ | = | 1 |
| 2³ | = | 8 |
| 3³ | = | 27 |
| 4³ | = | 64 |
| 5³ | = | 125 |
| 6³ | = | 216 |
| 7³ | = | 343 |
| 8³ | = | 512 |
| 9³ | = | 729 |
| 10³ | = | 1000 |
Powers & Roots Calculator Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
Powers in everyday life
Your TV screen resolution — 1920 × 1080 pixels — involves squaring and multiplication. A savings account paying 3% annual interest after 10 years uses 1.03¹⁰. The Richter scale measures earthquake magnitude logarithmically: an earthquake of magnitude 6 is 10 times more powerful than magnitude 5. Powers are how we describe growth, area, and intensity.
Worked example
"£500 is invested at 4% per year. How much is it worth after 6 years?"
Formula: A = P × (1 + r)ⁿ
A = 500 × (1.04)⁶
1.04⁶ = 1.2653…
Result: £632.66
The power (exponent) is the number of years. Without it, you'd have to multiply 1.04 six times by hand.
Key rules to remember
- → a⁰ = 1 always: Any number to the power zero equals 1. Even 1,000,000⁰ = 1.
- → Negative exponent = reciprocal: 2⁻³ = 1/2³ = 1/8 = 0.125
- → Fractional exponent = root: 8^(1/3) = ∛8 = 2
Powers vs multiplication
3⁴ means 3 × 3 × 3 × 3 = 81, NOT 3 × 4 = 12. The base is multiplied by itself, not by the exponent. This is the most common error with powers.
Understanding Powers and Roots
Powers (also called indices or exponents) are a way of showing repeated multiplication. For example, 2^3 means 2 multiplied by itself 3 times (2 × 2 × 2 = 8). The 'base' is the number being multiplied (2), and the 'exponent' is how many times it's multiplied (3).
Roots are the inverse operation of powers. For instance, if 2^3 = 8, then the cube root of 8 is 2. The most common roots are square roots (finding a number that, when multiplied by itself, gives the original number) and cube roots. Powers and roots are fundamental concepts in GCSE Maths, used in algebra, geometry, and problem-solving.
Laws of indices
A power indicates repeated multiplication of the base number by itself.
3^4 = 3 × 3 × 3 × 3 = 81
The exponent tells you how many times to multiply the base.
A negative exponent means you take the reciprocal of the base raised to the positive exponent.
5^-2 = 1 / 5^2 = 1 / 25
The negative sign in the exponent does NOT make the answer negative; it means 'one over'.
A fractional exponent with 1 in the numerator (e.g., 1/n) means you take the nth root of the base.
16^(1/2) = √16 = 4
The denominator of the fraction tells you which root to take.
For an exponent m/n, you can either take the nth root first and then raise to the power of m, or raise to the power of m first and then take the nth root.
8^(2/3) = (³√8)^2 = 2^2 = 4 OR 8^(2/3) = ³√(8^2) = ³√64 = 4
Taking the root first often makes calculations easier, especially without a calculator.
Any non-zero number raised to the power of zero is 1.
7^0 = 1
This is a special rule that applies to all numbers except 0^0, which is undefined.
Common mistakes to avoid
- -3^2 = 9:Without brackets, the power only applies to the 3, not the negative sign. So, -3^2 means -(3^2). (-3)^2 = (-3) × (-3) = 9. AND -3^2 = -(3 × 3) = -9.
- 5^0 = 0:A common misconception. Any non-zero number raised to the power of zero is 1. 5^0 = 1
- 9^(1/2) = 9 × (1/2) = 4.5:A fractional exponent means taking a root, not multiplying by the fraction. 9^(1/2) = √9 = 3
- √(9 + 16) = √9 + √16 = 3 + 4 = 7:You cannot split a square root over an addition or subtraction sign. √(9 + 16) = √25 = 5