Unit 8 · Lesson 1 of 6

Roots and simplifying radicals

About 25 minutes · 5 steps · one idea: pull the biggest perfect square (or cube) out of the root

You'll be able to: find square and cube roots and simplify radicals like \(\sqrt{72}\), \(\sqrt{50x^3}\), and \(\sqrt[3]{-54}\).

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first, then check.

Solve \(\dfrac{4}{x} = \dfrac{2}{x-3}\).

Excluded: \(x \ne 0\), \(x \ne 3\). Multiply both sides by \(x(x-3)\): \(4(x-3) = 2x\).

\(4x - 12 = 2x\), so \(2x = 12\) and \(x = 6\), which is allowed.

\((x^3)^2 = x^{n}\). What is \(n\)?

Power of a power: multiply the exponents. \(3 \cdot 2 = 6\), so \((x^3)^2 = x^6\) and \(n = 6\).

Step 2 of 5The idea

The square root \(\sqrt{a}\) is the nonnegative number whose square is \(a\): \(\sqrt{49} = 7\). The cube root \(\sqrt[3]{a}\) is the number whose cube is \(a\): \(\sqrt[3]{8} = 2\).

A cube root of a negative number is negative: \(\sqrt[3]{-8} = -2\), because \((-2)^3 = -8\). A square root of a negative number is not a real number (that comes later, in Complex numbers).

A radical is simplified when no perfect square (or cube, for cube roots) is left inside. Warm-up 2 is the reason variables work: \((x^3)^2 = x^6\), so \(\sqrt{x^6} = x^3\).

Split the root. \(\sqrt{ab} = \sqrt{a}\,\sqrt{b}\) and \(\sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}\).

Simplify a number. Write the radicand as (biggest perfect square) times (the rest). Take the root of the perfect square.

Perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100. Perfect cubes: 8, 27, 64, 125.

Simplify a variable (in this lesson, variables are never negative). Square root: halve the exponent, \(\sqrt{x^8} = x^4\). Cube root: divide the exponent by 3, \(\sqrt[3]{x^6} = x^2\). If it doesn't divide evenly, split off the leftover: \(\sqrt{x^5} = \sqrt{x^4}\,\sqrt{x} = x^2\sqrt{x}\).

Why this shows up on the placement test: answers with roots are always given in simplified form, so you need to recognize \(6\sqrt{2}\) as the same thing as \(\sqrt{72}\).

Step 3 of 5Worked examples

Example 1: \(\sqrt{72}\)

  1. Perfect squares that go into 72: 4, 9, and 36. The biggest is 36.
  2. \(\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36}\,\sqrt{2}\).
  3. \(\sqrt{36} = 6\), so the answer is \(6\sqrt{2}\). (Check: \((6\sqrt{2})^2 = 36 \cdot 2 = 72\).)

Example 2: \(\sqrt[3]{-54}\), then \(\sqrt{12x^5}\)

  1. For the cube root, find a perfect cube in 54: \(54 = 27 \cdot 2\). Keep the minus with the 27: \(-54 = -27 \cdot 2\).
  2. \(\sqrt[3]{-27} = -3\), so \(\sqrt[3]{-54} = -3\sqrt[3]{2}\).
  3. For \(\sqrt{12x^5}\): the biggest perfect square in 12 is 4, and the biggest even power in \(x^5\) is \(x^4\).
  4. \(\sqrt{12x^5} = \sqrt{4x^4}\,\sqrt{3x} = 2x^2\sqrt{3x}\). (Half of 4 is 2, so \(\sqrt{x^4} = x^2\).)

Step 4 of 5Practice

Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went. When a problem shows a box \(\square\), type only what goes in the box.

\(\sqrt{100} + \sqrt[3]{-8}\)

\(\sqrt{100} = 10\) and \(\sqrt[3]{-8} = -2\), because \((-2)^3 = -8\).

\(10 + (-2) = 8\).

Fill in the box: \(\sqrt{48} = \square\,\sqrt{3}\)

The biggest perfect square in 48 is 16: \(48 = 16 \cdot 3\).

\(\sqrt{48} = \sqrt{16}\,\sqrt{3} = 4\sqrt{3}\). The box is \(4\).

Simplify \(\sqrt{200}\) completely. The answer looks like \(a\sqrt{b}\). What number is \(a\), the number in front?

The biggest perfect square in 200 is 100: \(200 = 100 \cdot 2\).

\(\sqrt{200} = \sqrt{100}\,\sqrt{2} = 10\sqrt{2}\), so \(a = 10\). (If you got \(2\sqrt{50}\) or \(5\sqrt{8}\), keep going: both still have a perfect square inside.)

Fill in the box: \(\sqrt[3]{-250} = \square\,\sqrt[3]{2}\)

Look for a perfect cube in 250: \(250 = 125 \cdot 2\). Keep the minus with the 125: \(-250 = -125 \cdot 2\).

\(\sqrt[3]{-125} = -5\), so \(\sqrt[3]{-250} = -5\sqrt[3]{2}\). The box is \(-5\).

Which is the simplified form of \(\sqrt{49x^2y^6}\)? (Assume \(x\) and \(y\) are not negative.)

  1. \(49xy^3\)
  2. \(7xy^2\)
  3. \(7xy^3\)
  4. \(7x^2y^6\)

Take each root separately: \(\sqrt{49} = 7\), \(\sqrt{x^2} = x\), \(\sqrt{y^6} = y^3\) (half of 6 is 3). Answer: \(7xy^3\). Choice C.

(A) forgot the root of 49. (B) divided the exponent 6 by 3; for a square root you halve it. (D) took the root of 49 only.

Fill in the box: \(\sqrt{50x^3} = \square\,\sqrt{2x}\). (Assume \(x\) is not negative.)

Biggest perfect square in 50: 25. Biggest even power in \(x^3\): \(x^2\).

\(\sqrt{50x^3} = \sqrt{25x^2}\,\sqrt{2x} = 5x\sqrt{2x}\). The box is \(5x\).

Fill in the box: \(\sqrt[3]{16x^5} = \square\,\sqrt[3]{2x^2}\). (Assume \(x\) is not negative.)

Biggest perfect cube in 16: 8. Biggest power of \(x\) in \(x^5\) with an exponent divisible by 3: \(x^3\).

\(\sqrt[3]{16x^5} = \sqrt[3]{8x^3}\,\sqrt[3]{2x^2} = 2x\sqrt[3]{2x^2}\). The box is \(2x\).

Which is the simplified form of \(\sqrt{18x^5}\)? (Assume \(x\) is not negative.)

  1. \(9x^2\sqrt{2x}\)
  2. \(3x^2\sqrt{2x}\)
  3. \(2x^2\sqrt{3x}\)
  4. \(3x^4\sqrt{2x}\)

\(18x^5 = 9x^4 \cdot 2x\). Then \(\sqrt{9x^4} = 3x^2\), and \(2x\) stays inside: \(3x^2\sqrt{2x}\). Choice B.

(A) forgot to take the root of 9. (C) mixed up which number came out. (D) forgot to halve the exponent 4.

Step 5 of 5Wrap-up

You can now find square and cube roots and simplify radicals: pull out the biggest perfect square (or cube), halve exponents under a square root, and divide them by 3 under a cube root.

Read or watch next: OpenStax Intermediate Algebra 2e, 8.1 Simplify Expressions with Roots and 8.2 Simplify Radical Expressions (free textbook, with more practice at the end of each section).

Also good: Khan Academy: Simplifying square roots (short videos). Quick rules: Radicals and complex numbers cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "How do I find the biggest perfect square inside a number quickly?" It knows this course and can explain things another way.