Radicals: Meaning, Rules, Simplifying, Operations, and Equations

What is a radical?

A radical is an expression containing a root. Roots undo powers in the same way that division undoes multiplication.

The square root \(\sqrt{25}=5\) because \(5^2=25\). The cube root \(\sqrt[3]{27}=3\) because \(3^3=27\).

A general radical is written as \(\sqrt[n]{a}\). It asks for a number whose \(n\)-th power equals \(a\). In symbols, \(\sqrt[n]{a}=b\) means \(b^n=a\).

For focused work with third powers and third roots, see Cubes and Cube Roots.

Parts of a radical expression

In \(\sqrt[n]{a}\), the symbol \(\sqrt{\phantom{a}}\) is the radical sign, \(n\) is the index, and \(a\) is the radicand. The entire expression is called a radical expression.

If no index is written, the index is understood to be \(2\). Therefore, \(\sqrt{a}=\sqrt[2]{a}\).

In \(4\sqrt{7}\), the number \(4\) outside the radical is the coefficient. It means \(4\times\sqrt{7}\).

The index must be a whole number greater than \(1\). The radicand may be a number, variable, or algebraic expression.

Square roots and the principal root

The symbol \(\sqrt{a}\) means the principal square root, which is the nonnegative square root.

For example, \(\sqrt{49}=7\), not \(\pm7\). However, the equation \(x^2=49\) has two solutions: \(x=7\) and \(x=-7\), written \(x=\pm7\).

This distinction is important: \(\sqrt{49}\) is one nonnegative value, while solving \(x^2=49\) asks for every number whose square is \(49\).

Also, \(\sqrt{x^2}=|x|\), not always \(x\). For example, if \(x=-5\), then \(\sqrt{x^2}=\sqrt{25}=5=|-5|\).

Perfect squares and perfect roots

A perfect square is the square of an integer. Recognizing perfect squares makes radical simplification much faster.

What does simplest radical form mean?

A radical is in simplest radical form when the radicand contains no perfect-power factor matching the index, there is no fraction inside the radical, and no radical remains in the denominator.

For a square root, remove every perfect-square factor. For example, \(\sqrt{12}\) is not simplified because \(12=4\times3\). Its simplified form is \(2\sqrt{3}\).

The expressions \(\sqrt{12}\) and \(2\sqrt{3}\) have the same value because \(\sqrt{12}=\sqrt{4\times3}=\sqrt4\sqrt3=2\sqrt3\).

Method 1 (easiest): use the largest perfect-square factor

This is usually the fastest method for simplifying a numerical square root.

Step-by-step example: Simplify \(\sqrt{72}\).

Step 1: Find the largest perfect-square factor of \(72\). Since \(72=36\times2\), use \(36\).

Step 2: Split the product: \(\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\sqrt2\).

Step 3: Evaluate the perfect root: \(\sqrt{36}=6\).

Answer: \(\sqrt{72}=6\sqrt2\).

Check: \((6\sqrt2)^2=6^2(\sqrt2)^2=36\times2=72\).

Method 2: use prime-factor pairs

Prime factorization is systematic and works well when the largest perfect-square factor is not obvious. For a square root, every pair of equal factors produces one factor outside the radical.

Step-by-step example: Simplify \(\sqrt{180}\).

Step 1: Prime-factorize: \(180=2\times2\times3\times3\times5\).

Step 2: Group equal pairs: \(180=(2\times2)(3\times3)\times5\).

Step 3: Take one factor from each pair outside: \(\sqrt{180}=2\times3\sqrt5\).

Answer: \(\sqrt{180}=6\sqrt5\).

The unpaired factor \(5\) remains inside the radical.

Product and quotient rules

The product rule combines or separates multiplication under square roots: \(\sqrt{ab}=\sqrt a\sqrt b\), provided \(a\ge0\) and \(b\ge0\).

The quotient rule combines or separates division: \(\sqrt{\frac{a}{b}}=\frac{\sqrt a}{\sqrt b}\), provided \(a\ge0\) and \(b\gt0\).

Product example: \(\sqrt{12}=\sqrt{4\times3}=2\sqrt3\).

Quotient example: \(\sqrt{\frac{49}{64}}=\frac{\sqrt{49}}{\sqrt{64}}=\frac78\).

These rules apply to products and quotients, not to addition or subtraction. In general, \(\sqrt{a+b}\ne\sqrt a+\sqrt b\).

Radical rules at a glance

Read the explanation column before applying a symbolic rule. Every condition matters.

Add and subtract like radicals

Radicals are like radicals when they have the same index and the same simplified radicand. Combine their coefficients just as you combine like algebraic terms.

Example 1: \(3\sqrt5+4\sqrt5=(3+4)\sqrt5=7\sqrt5\).

Example 2: \(9\sqrt2-5\sqrt2=(9-5)\sqrt2=4\sqrt2\).

Simplify before deciding whether radicals are alike. For example, \(\sqrt8+\sqrt{18}=2\sqrt2+3\sqrt2=5\sqrt2\).

The expression \(\sqrt2+\sqrt3\) cannot be combined because the radicands are different.

Multiply radical expressions

Multiply coefficients, multiply radical parts, and then simplify.

Step-by-step example: Multiply \((2\sqrt6)(3\sqrt{15})\).

Step 1: Multiply coefficients: \(2\times3=6\).

Step 2: Multiply radicands: \(\sqrt6\sqrt{15}=\sqrt{90}\).

Step 3: Simplify \(\sqrt{90}=\sqrt{9\times10}=3\sqrt{10}\).

Step 4: Multiply the outside factors: \(6\times3\sqrt{10}=18\sqrt{10}\).

Answer: \((2\sqrt6)(3\sqrt{15})=18\sqrt{10}\).

Distribute and multiply radical binomials

Use the distributive property exactly as you would with variables.

Example 1: \(\sqrt3(2+\sqrt3)=2\sqrt3+(\sqrt3)^2=2\sqrt3+3\).

Example 2: \((\sqrt5+2)(\sqrt5-2)\). These are conjugates, so the middle terms cancel.

\[(\sqrt5+2)(\sqrt5-2)=(\sqrt5)^2-2^2=5-4=1\]

The identity \((a+b)(a-b)=a^2-b^2\) makes conjugate products especially useful.

Divide radicals and rationalize a denominator

A simplified exact answer traditionally has no radical in its denominator. Rationalizing means multiplying by a form of \(1\) that removes the denominator radical.

Step-by-step example: Simplify \(\frac{5}{\sqrt3}\).

Step 1: Multiply by \(\frac{\sqrt3}{\sqrt3}\), which equals \(1\).

\[\frac{5}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{5\sqrt3}{(\sqrt3)^2}\]

Step 2: Since \((\sqrt3)^2=3\), simplify the denominator.

Answer: \(\frac{5}{\sqrt3}=\frac{5\sqrt3}{3}\).

Fraction example: \(\sqrt{\frac{3}{8}}=\frac{\sqrt3}{2\sqrt2}=\frac{\sqrt6}{4}\).

Advanced extension: rationalize with a conjugate

When the denominator has two terms and one contains a square root, multiply by its conjugate. The conjugate changes the sign between the terms.

Example: Rationalize \(\frac{1}{2+\sqrt3}\). The conjugate of \(2+\sqrt3\) is \(2-\sqrt3\).

\[\frac{1}{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}=\frac{2-\sqrt3}{2^2-(\sqrt3)^2}\]

The denominator becomes \(4-3=1\). Therefore, \(\frac{1}{2+\sqrt3}=2-\sqrt3\).

Higher roots and real-number restrictions

An even-index root, such as a square root or fourth root, has a real value only when its radicand is nonnegative.

For real numbers, \(\sqrt{16}=4\), but \(\sqrt{-16}\) is not real. Likewise, \(\sqrt[4]{81}=3\).

An odd-index root can contain any real radicand. For example, \(\sqrt[3]{-125}=-5\) because \((-5)^3=-125\).

To simplify an \(n\)-th root, take out groups of \(n\) equal factors. For example, \(\sqrt[3]{54}=\sqrt[3]{27\times2}=3\sqrt[3]{2}\).

Radicals and rational exponents

A radical can be rewritten with a fractional exponent:

\[\sqrt[n]{a}=a^{1/n}\]

More generally, \(a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m\), whenever the expression is defined in the number system being used.

Example 1: \(16^{1/2}=\sqrt{16}=4\).

Example 2: \(27^{2/3}=(\sqrt[3]{27})^2=3^2=9\).

Example 3: \(81^{3/4}=(\sqrt[4]{81})^3=3^3=27\).

Use the Exponent Calculator to check rational-exponent evaluations after rewriting them by hand.

Simplify radicals containing variables

Factor variable powers into perfect groups that match the radical index.

Example: Simplify \(\sqrt{72x^5}\) for \(x\ge0\).

Step 1: Factor the numerical part: \(72=36\times2\).

Step 2: Factor the variable part: \(x^5=x^4\times x\).

Step 3: Separate perfect squares: \(\sqrt{72x^5}=\sqrt{36x^4\times2x}\).

Step 4: Take the perfect squares outside: \(\sqrt{36}=6\) and \(\sqrt{x^4}=x^2\).

Answer: \(\sqrt{72x^5}=6x^2\sqrt{2x}\) for \(x\ge0\).

Find the domain of a radical expression

For a real-valued even root, require the radicand to be at least \(0\).

Example: Find the domain of \(\sqrt{5-2x}\).

Step 1: Require \(5-2x\ge0\).

Step 2: Subtract \(5\): \(-2x\ge-5\).

Step 3: Divide by \(-2\) and reverse the inequality: \(x\le\frac52\).

Answer: The real domain is \(x\le\frac52\), or \(( -\infty,\frac52 ]\).

If a radical is in a denominator, the radicand must make the denominator real and nonzero. For example, \(\frac{1}{\sqrt{x-3}}\) requires \(x\gt3\).

Solve a basic radical equation

Isolate the radical, raise both sides to the matching power, solve, and then check the answer in the original equation.

Step-by-step example: Solve \(\sqrt{2x-1}=5\).

Step 1: The radical is already isolated.

Step 2: Square both sides: \((\sqrt{2x-1})^2=5^2\).

Step 3: Simplify: \(2x-1=25\).

Step 4: Solve: \(2x=26\), so \(x=13\).

Step 5: Check: \(\sqrt{2(13)-1}=\sqrt{25}=5\).

Answer: \(x=13\).

Extraneous solutions must be checked

Squaring can create a solution that does not satisfy the original equation. Such an answer is called an extraneous solution.

Example: Solve \(\sqrt{x+1}=x-1\).

Step 1: Because a principal square root is nonnegative, the right side must satisfy \(x-1\ge0\), so \(x\ge1\).

Step 2: Square both sides: \(x+1=(x-1)^2=x^2-2x+1\).

Step 3: Rearrange: \(x^2-3x=0\), so \(x(x-3)=0\). The candidates are \(x=0\) and \(x=3\).

Step 4: Check \(x=0\): \(\sqrt1\ne-1\), so \(0\) is extraneous.

Step 5: Check \(x=3\): \(\sqrt4=2\) and \(3-1=2\), so \(3\) works.

Answer: \(x=3\).

Estimate and approximate radicals

When a radical does not simplify to a whole number, locate it between nearby perfect roots.

Example: Estimate \(\sqrt{50}\). Since \(7^2=49\) and \(8^2=64\), \(7\lt\sqrt{50}\lt8\). Because \(50\) is only slightly greater than \(49\), the value is slightly greater than \(7\).

A calculator gives \(\sqrt{50}\approx7.071\). The exact simplified form is \(5\sqrt2\).

Use \(=\) for an exact radical form and \(\approx\) for a rounded decimal. The Root Calculator can check square, cube, and higher-root approximations.

Geometry applications

Radicals appear whenever a length is recovered from an area or from the Pythagorean theorem.

Square example: A square has area \(72\text{ cm}^2\). If its side length is \(s\), then \(s^2=72\), so \(s=\sqrt{72}=6\sqrt2\text{ cm}\).

Right-triangle example: A right triangle has legs \(5\) and \(8\). Its hypotenuse \(c\) satisfies \(c^2=5^2+8^2=89\), so \(c=\sqrt{89}\approx9.434\).

The exact answer preserves the radical. Use a decimal only when measurement or the question requires an approximation.

Common mistakes

Do not split addition or subtraction: \(\sqrt{a+b}\ne\sqrt a+\sqrt b\) in general. For example, \(\sqrt{9+16}=5\), but \(\sqrt9+\sqrt{16}=7\).

Do not forget the principal-root convention. \(\sqrt{36}=6\), while solving \(x^2=36\) gives \(x=\pm6\).

Do not combine unlike radicals. The expression \(2\sqrt3+4\sqrt5\) cannot be simplified by addition.

Do not combine radicals before checking the index. The expressions \(\sqrt2\) and \(\sqrt[3]{2}\) are unlike.

Do not leave perfect-square factors inside. Write \(3\sqrt2\) instead of \(\sqrt{18}\) when simplest radical form is requested.

Do not divide an inequality by a negative number without reversing its direction.

Do not accept a radical-equation solution before substituting it into the original equation.

Practice questions

1. Evaluate \(\sqrt{121}\).

2. Evaluate \(\sqrt[3]{-64}\).

3. Simplify \(\sqrt{48}\).

4. Simplify \(\sqrt{200}\).

5. Simplify \(\sqrt[3]{128}\).

6. Add \(3\sqrt7+5\sqrt7\).

7. Simplify \(\sqrt{12}+\sqrt{27}\).

8. Multiply \(2\sqrt3\times4\sqrt6\).

9. Expand \(\sqrt2(3+\sqrt2)\).

10. Rationalize \(\frac{4}{\sqrt5}\).

11. Evaluate \(64^{2/3}\).

12. Simplify \(\sqrt{50x^3}\) for \(x\ge0\).

13. Find the real domain of \(\sqrt{12-3x}\).

14. Solve \(\sqrt{x+4}=6\).

15. Solve and check \(\sqrt{x+6}=x\).

16. Place \(\sqrt{30}\) between two consecutive whole numbers.

17. A square has area \(98\text{ m}^2\). Find its exact side length.

18. Explain why \(\sqrt{4+5}\ne\sqrt4+\sqrt5\).

Practice answers

1. \(\sqrt{121}=11\).

2. \(\sqrt[3]{-64}=-4\).

3. \(\sqrt{48}=4\sqrt3\).

4. \(\sqrt{200}=10\sqrt2\).

5. \(\sqrt[3]{128}=4\sqrt[3]{2}\).

6. \(3\sqrt7+5\sqrt7=8\sqrt7\).

7. \(\sqrt{12}+\sqrt{27}=2\sqrt3+3\sqrt3=5\sqrt3\).

8. \(2\sqrt3\times4\sqrt6=8\sqrt{18}=24\sqrt2\).

9. \(\sqrt2(3+\sqrt2)=3\sqrt2+2\).

10. \(\frac{4}{\sqrt5}=\frac{4\sqrt5}{5}\).

11. \(64^{2/3}=(\sqrt[3]{64})^2=4^2=16\).

12. \(\sqrt{50x^3}=5x\sqrt{2x}\) for \(x\ge0\).

13. \(12-3x\ge0\), so \(x\le4\).

14. \(x+4=36\), so \(x=32\).

15. Squaring gives \(x+6=x^2\), so \((x-3)(x+2)=0\). Checking leaves \(x=3\); \(x=-2\) is extraneous.

16. Since \(25\lt30\lt36\), \(5\lt\sqrt{30}\lt6\).

17. \(s=\sqrt{98}=7\sqrt2\text{ m}\).

18. \(\sqrt{4+5}=\sqrt9=3\), while \(\sqrt4+\sqrt5=2+\sqrt5\), and these values are not equal.

The complete radical strategy

First identify the index and radicand. Check real-number restrictions, then simplify by removing the largest perfect-power factor or by grouping prime factors.

For operations, simplify first. Combine only like radicals, multiply using the product rule and distributive property, and rationalize a denominator when exact simplest form is required.

For equations, isolate the radical, raise both sides to the matching power, solve, and always check for extraneous solutions. Keep exact radical form unless the question asks for a decimal approximation.