Square root

The square root of a number is the value that gives that number when you multiply it by itself. The square root of 25 is 5, because 5 × 5 = 25, and it is written √25 = 5. Every positive number has two square roots, one positive and one negative (5 and −5), but the √ symbol always means the positive one.

What is a square root?

Squaring a number means multiplying it by itself: 7² = 7 × 7 = 49. Taking the square root goes backwards: √49 = 7. The two operations undo each other. The name comes from squares: a square with an area of 49 cm² has sides of √49 = 7 cm.

The symbol √ is called the radical sign, and the number under it is the radicand. In √49, the radicand is 49. A number like 49, whose square root is a whole number, is called a perfect square.

Square root table

The square roots of numbers that are not perfect squares, such as √2, go on forever without repeating. They are irrational numbers: they can't be written as a fraction. The US Common Core expects grade 8 students to know that √2 is irrational, and to evaluate square roots of small perfect squares. In England, AQA's GCSE Maths specification expects students to know square numbers up to 15 × 15.

Positive and negative square roots

Both 6 × 6 and (−6) × (−6) equal 36, so 36 has two square roots: 6 and −6. The symbol √36 means only the positive one, 6, called the principal square root. When you solve an equation, though, you need both:

The ± sign means "plus or minus", so x = 6 or x = −6. Forgetting the negative answer is one of the most common slips in algebra.

How to find a square root

1. Know the perfect squares

Learn the table above and the common ones are instant. It also helps with numbers ending in zeros: √3,600 = 60, because 60 × 60 = 3,600.

2. Use prime factors

Write the number as a product of primes, then take half of each power. For 324 = 2² × 3⁴, the square root is 2 × 3² = 18. Check: 18 × 18 = 324. The prime factorisation calculator splits any number into primes.

3. Estimate between perfect squares

50 is between 49 and 64, so √50 is between 7 and 8, and much closer to 7. Try 7.1: 7.1² = 50.41, a little too big. So √50 is just under 7.1. A calculator gives 7.071.

4. Improve a guess

For a more accurate answer by hand, take a guess, divide the number by it, and average the two. For √10, start with 3. Then 10 ÷ 3 = 3.333, and the average of 3 and 3.333 is 3.1667. Repeat with 3.1667 and you get 3.1623, which is already right to 4 decimal places. This is often called the Babylonian method.

Simplifying square roots

A square root that isn't a whole number can often be written more simply. In the UK these are called surds, and in the US, simplifying radicals. The key rule is that you can split a square root over multiplication:

To simplify, find the biggest perfect square that divides the number:

• √72 = √(36 × 2) = √36 × √2 = 6√2
• √48 = √(16 × 3) = 4√3
• √200 = √(100 × 2) = 10√2

The exponent calculator simplifies any square root as a surd and shows the working.

Square roots of fractions and decimals

For a fraction, take the square root of the top and the bottom: √(9/16) = 3/4. For a decimal, think of the matching whole number: √0.25 = 0.5 because 0.5 × 0.5 = 0.25, and √1.44 = 1.2. Notice that the square root of a number between 0 and 1 is bigger than the number itself: √0.09 = 0.3.

Square roots as powers

A square root is the same as raising to the power of one half:

This fits the index laws: x^½ × x^½ = x^1 = x, just as √x × √x = x. So 16^½ = 4 and 100^½ = 10. Cube roots work the same way with a power of one third.

Square root of a negative number

No real number squares to a negative, because positive × positive and negative × negative are both positive. So √(−9) has no real answer. In more advanced maths, mathematicians define i as √(−1), and then √(−9) = 3i, an imaginary number. The complex numbers calculator works with these.

The square root graph

Here x is the number and y is its square root. The curve starts at (0, 0), rises quickly and then flattens: to double the square root you need four times the number. There is nothing to the left of zero, because negative numbers have no real square root. The Graphing Functions course has a section on square root functions and how to shift them.

Worked examples

Example 1: the side of a square

A square garden has an area of 81 m². Its side is √81 = 9 m. Only the positive root makes sense, because a side is never shorter than zero.

Example 2: Pythagoras' theorem

A right-angled triangle (right triangle in the US) has shorter sides of 6 cm and 8 cm. The hypotenuse is:

The hypotenuse is 10 cm. Square first, add, then take the square root of the total, never of each part.

Example 3: solving an equation

Solve 3x² = 48. Divide by 3 to get x² = 16, then take the square root of both sides: x = 4 or −4.

Common mistakes

• Halving instead of rooting. √16 is 4, not 8.
• Forgetting ± when solving. x² = 25 has two answers, 5 and −5.
• Splitting over addition. √(a + b) is not √a + √b.
• Not simplifying fully. √72 = 2√18 is true but not finished: the answer is 6√2.
• Thinking roots always make numbers smaller. For numbers between 0 and 1, the square root is bigger.

To practise roots among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 paper. In the UK, the same practice opens as GCSE Maths Foundation.

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