√72
36 × 2 = 72. √36 = 6. √72 = 6√2 ≈ 8.485.
Square root and exponent calculator with steps
What do you want?
√72
8.485281
Exact: 6√2
How we got this
- Find the largest perfect square that divides 72: 6² = 36.
- 72 = 36 × 2.
- √72 = √36 × √2 = 6√2 ≈ 8.485281.
Exact answers first
Exams usually want √72 as 6√2, not 8.485. To get there, find the largest perfect square that divides the number, take its root outside, and leave the rest under the sign. Exponents follow three rules worth memorising: multiply powers by adding exponents, a negative exponent is a reciprocal, and a fractional exponent is a root.
Very large and small powers are easier to read in scientific notation. The quadratic formula uses a square root at its core; see the quadratic equation solver. Roots of fractions are roots of top and bottom, which the fraction calculator can then simplify.
Sources
What this calculator does not account for
- Complex numbers: square roots of negative numbers are not shown.
- Rounding beyond the decimals shown.
How this is calculated
Formula: Roots as powers
√(a²b) =a√b; ⁿ√x =x^(1/n); x^(−n) =1 ÷ xⁿWhat the symbols mean
- a²The largest perfect square that divides the number.
Good to know
- An odd root of a negative number is negative. An even root of a negative number is not a real number.
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Worked examples
Each example is worked out by the same code as the calculator above.
8^(2/3)
The cube root of 8 is 2, and squared that is 4.
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<iframe src="/embed/square-root-exponent" title="Roots and exponents" width="100%" height="720" style="border:0;max-width:42rem" loading="lazy"></iframe>Questions
How do I simplify √72?
72 = 36 × 2, so √72 = 6√2 ≈ 8.485.
What is a negative exponent?
One over the positive power: 2^−3 = 1/8 = 0.125.
What is the cube root of 27?
3, because 3 × 3 × 3 = 27.