Simplifying Square Roots

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Simplifying square roots means rewriting a square root so no perfect square factor is left under the radical sign. Find the largest perfect square that divides the number, split it out, and take its square root. For example, \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}. This is called simplest radical form.

Before this lesson: You should know square roots and perfect squares, prime factorization and the exponent rules.

What Does It Mean to Simplify a Square Root?

Some square roots are whole numbers, like \sqrt{36} = 6. Others are not, like \sqrt{72}. Its value is a decimal that never ends or repeats, so it is an irrational number.

You can still write \sqrt{72} in a simpler, exact way. The number 72 has a perfect square hiding inside it: 72 = 36 \cdot 2. The square root of 36 is 6, so 6 can come out in front.

The number under the radical sign is the radicand. A square root is in simplest radical form when the radicand has no perfect square factor other than 1.

Three rows of steps. Top: √72. An arrow labeled find the largest perfect square factor, 36, points to the middle row: √(36 · 2), with 36 in a blue box. An arrow labeled √36 = 6, so 6 comes out, points to the bottom row: 6√2, with 6 in a blue box.
Split 72 into 36 · 2. The square root of 36 is 6, and the 2 stays under the radical: √72 = 6√2.

The Product Property of Square Roots

\sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \quad \text{for } a \ge 0 \text{ and } b \ge 0

This rule lets you split one square root into two. Pick a to be a perfect square. Then \sqrt{a} is a whole number.

Perfect squares to look forTip
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Try the largest one that divides the radicand.

Steps for simplifying a square root

  1. Find the largest perfect square that divides the radicand.
  2. Write the radicand as that perfect square times another number.
  3. Use \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} to split the root.
  4. Take the square root of the perfect square. Write it in front.
  5. Check that the number left under the radical has no perfect square factor.

Worked Examples

Example 1: A basic square root

Simplify \sqrt{50}.

The largest perfect square that divides 50 is 25.

\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}

Check: (5\sqrt{2})^2 = 25 \cdot 2 = 50.

Example 2: When you don’t pick the largest square

Simplify \sqrt{72}. Suppose you see 4 first instead of 36.

\begin{aligned} \sqrt{72} &= \sqrt{4 \cdot 18} = 2\sqrt{18} \\ &= 2\sqrt{9 \cdot 2} = 2 \cdot 3\sqrt{2} \\ &= 6\sqrt{2} \end{aligned}

You get the same answer, 6\sqrt{2}, but it takes more steps. The radicand 18 still had a perfect square factor, so you had to keep going.

Example 3: A number already in front

Simplify 3\sqrt{20}.

Simplify the root first. Then multiply the numbers in front.

3\sqrt{20} = 3\sqrt{4 \cdot 5} = 3 \cdot 2\sqrt{5} = 6\sqrt{5}

Example 4: Square roots with variables

Simplify \sqrt{18x^2} and \sqrt{x^5}. Assume x \ge 0.

An even power is a perfect square: x^2 = (x)^2 and x^4 = (x^2)^2. For an odd power, split off one x.

\begin{aligned} \sqrt{18x^2} &= \sqrt{9 \cdot 2 \cdot x^2} = 3x\sqrt{2} \\ \sqrt{x^5} &= \sqrt{x^4 \cdot x} = x^2\sqrt{x} \end{aligned}

Example 5: Word problem

A square patio has an area of 48 square feet. Find the exact length of one side in simplest radical form. Then estimate it, using \sqrt{3} \approx 1.73.

The side of a square is the square root of its area. The largest perfect square that divides 48 is 16.

\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}

Each side is 4\sqrt{3} feet. That is about 4 \cdot 1.73 = 6.92 feet, or a little less than 7 feet.

Common Mistakes

MistakeExample of the errorFix
Splitting a sum\sqrt{9 + 16} = 3 + 4 = 7The product rule only works for products. \sqrt{9 + 16} = \sqrt{25} = 5.
Stopping too soon\sqrt{72} = 2\sqrt{18}18 still has the factor 9. Keep going: 6\sqrt{2}.
Putting the square in front\sqrt{50} = 25\sqrt{2}Write the square root of 25 in front, not 25: 5\sqrt{2}.
Dropping the number in front3\sqrt{20} = 2\sqrt{5}Multiply by the 3 too: 3 \cdot 2\sqrt{5} = 6\sqrt{5}.
Taking half of the radicand\sqrt{16} = 8\sqrt{16} = 4, because 4 \cdot 4 = 16.

Other Ways to Simplify

Using prime factorization

If you can’t spot the perfect square, break the radicand into primes. Each pair of the same prime comes out as one number.

\sqrt{180} = \sqrt{2 \cdot 2 \cdot 3 \cdot 3 \cdot 5} = 2 \cdot 3\sqrt{5} = 6\sqrt{5}

Square roots of fractions

When the bottom of a fraction is a perfect square, take the root of the top and the bottom. For example, \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}. Roots on the bottom that are not perfect squares come later, in rationalizing the denominator.

Simplified or not?

Square rootSimplified?Why
\sqrt{30}Yes30 = 2 \cdot 3 \cdot 5 has no perfect square factor
\sqrt{28}No28 = 4 \cdot 7, so \sqrt{28} = 2\sqrt{7}
5\sqrt{3}Yes3 has no perfect square factor
\sqrt{x^3}No\sqrt{x^3} = x\sqrt{x} for x \ge 0

Like terms with the same radical can be added and subtracted, much like combining like terms. That is the next skill in this unit, Adding and Subtracting Radicals.

Practice Problems

  1. Simplify \sqrt{12}.
  2. Simplify \sqrt{45}.
  3. Simplify \sqrt{98}.
  4. Simplify \sqrt{200}.
  5. Simplify \sqrt{75}.
  6. Simplify 2\sqrt{27}.
  7. Simplify \sqrt{49x^2}. Assume x \ge 0.
  8. Simplify \sqrt{12x^3}. Assume x \ge 0.
  9. Simplify \sqrt{\frac{11}{36}}.
  10. A square tile has an area of 32 square inches. Find the exact length of one side. Then estimate it, using \sqrt{2} \approx 1.41.
Show answers
  1. \sqrt{4 \cdot 3} = 2\sqrt{3}
  2. \sqrt{9 \cdot 5} = 3\sqrt{5}
  3. \sqrt{49 \cdot 2} = 7\sqrt{2}
  4. \sqrt{100 \cdot 2} = 10\sqrt{2}
  5. \sqrt{25 \cdot 3} = 5\sqrt{3}
  6. 2\sqrt{9 \cdot 3} = 2 \cdot 3\sqrt{3} = 6\sqrt{3}
  7. 7x. Both 49 and x^2 are perfect squares.
  8. \sqrt{4x^2 \cdot 3x} = 2x\sqrt{3x}
  9. \frac{\sqrt{11}}{6}. 11 has no perfect square factor.
  10. \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} inches. That is about 4 \cdot 1.41 = 5.64 inches.

FAQ

What is simplest radical form?

A square root is in simplest radical form when the number under the radical has no perfect square factor other than 1. For example, 6\sqrt{2} is in simplest form, but \sqrt{72} is not.

How do you simplify a square root that is not a perfect square?

Find the largest perfect square that divides the number. Split the root into that square times the rest, and take the square root of the square.

Can you split the square root of a sum?

No. \sqrt{a + b} is not \sqrt{a} + \sqrt{b}. Add first, then take the root.

Is the square root of 8 the same as 2 root 2?

Yes. \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}. They have the same value, about 2.83, but 2\sqrt{2} is the simplified form.

Summary

  • Use \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} to split a square root into two.
  • Find the largest perfect square factor, take its root, and write it in front.
  • If the number left still has a perfect square factor, keep going.
  • Even powers of a variable are perfect squares: \sqrt{x^4} = x^2 when x \ge 0.
  • Never split the square root of a sum.

Previous lesson: Factoring Completely

Next lesson: Adding and Subtracting Radicals

Back to the Algebra 1 Index

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