maths.free › Algebra › 8. Roots and Radicals › Add, Subtract, and Multiply Radical Expressions

Add, Subtract, and Multiply Radical Expressions

Add and subtract radical expressions

Add and Subtract Radical Expressions

Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.

We add and subtract like radicals in the same way we add and subtract like terms. We know that \(3x+8x\) is \(11x.\) Similarly we add \(3\sqrt{x}+8\sqrt{x}\) and the result is \(11\sqrt{x}.\)

Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.

Example

Try it.

Simplify: ⓐ \(2\ \sqrt{2}-7\ \sqrt{2}\) ⓑ \(5\ \sqrt[3]{y}+4\ \sqrt[3]{y}\) ⓒ \(7\ \sqrt[4]{x}-2\ \sqrt[4]{y}.\)

Solution

ⓐ

	\(\ 2\ \sqrt{2}-7\ \sqrt{2}\)
Since the radicals are like, we subtract the coefficients.	\(\ -5\ \sqrt{2}\)

ⓑ

	\(\ 5\ \sqrt[3]{y}+4\ \sqrt[3]{y}\)
Since the radicals are like, we add the coefficients.	\(\ 9\ \sqrt[3]{y}\)

ⓒ

\(\ 7\ \sqrt[4]{x}-2\ \sqrt[4]{y}\)

The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.

For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.

Example

Try it.

Simplify: ⓐ \(2\ \sqrt{5n}-6\ \sqrt{5n}+4\ \sqrt{5n}\) ⓑ \(\sqrt[4]{3xy}+5\ \sqrt[4]{3xy}-4\ \sqrt[4]{3xy}.\)

Solution

ⓐ

	\(\ 2\ \sqrt{5n}-6\ \sqrt{5n}+4\ \sqrt{5n}\)
Since the radicals are like, we combine them.	\(\ 0\ \sqrt{5n}\)
Simplify.	\(\ 0\)

ⓑ

	\(\ \sqrt[4]{3xy}+5\ \sqrt[4]{3xy}-4\ \sqrt[4]{3xy}\)
Since the radicals are like, we combine them.	\(\ 2\ \sqrt[4]{3xy}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Multiply Radical Expressions

We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Roots so we see both ways together.

When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply \(4x\cdot 3y\) we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.

Example

Try it.

Simplify: ⓐ \((6\ \sqrt{2})(3\ \sqrt{10})\) ⓑ \((-5\ \sqrt[3]{4})(-4\ \sqrt[3]{6}).\)

Solution

ⓐ

	\(\ (6\ \sqrt{2})(3\ \sqrt{10})\)
Multiply using the Product Property.	\(\ 18\ \sqrt{20}\)
Simplify the radical.	\(\ 18\ \sqrt{4}\cdot \sqrt{5}\)
Simplify.	\(\ 18\cdot 2\cdot \sqrt{5}\)
	\(\ 36\ \sqrt{5}\)

ⓑ

	\(\ (-5\ \sqrt[3]{4})(-4\ \sqrt[3]{6})\)
Multiply using the Product Property.	\(\ 20\ \sqrt[3]{24}\)
Simplify the radical.	\(\ 20\ \sqrt[3]{8}\cdot \sqrt[3]{3}\)
Simplify.	\(\ 20\cdot 2\cdot \sqrt[3]{3}\)
	\(\ 40\ \sqrt[3]{3}\)

We follow the same procedures when there are variables in the radicands.

Example

Try it.

Simplify: ⓐ \((10\ \sqrt{6{p}^{3}})(4\ \sqrt{3p})\) ⓑ \((2\ \sqrt[4]{20{y}^{2}})(3\ \sqrt[4]{28{y}^{3}}).\)

Solution

ⓐ

	\(\ (10\sqrt{6{p}^{3}})(4\sqrt{3p})\)
Multiply.	\(\ 40\sqrt{18{p}^{4}}\)
Simplify the radical.	\(\ 40\sqrt{9{p}^{4}}\cdot \sqrt{2}\)
Simplify.	\(\ 40\cdot 3{p}^{2}\cdot \sqrt{2}\)
	\(\ 120{p}^{2}\sqrt{2}\)

ⓑ When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.

	\(\ (2\sqrt[4]{20{y}^{2}})(3\sqrt[4]{28{y}^{3}})\)
Multiply.	\(\ 6\ \sqrt[4]{4\cdot 5\cdot 4\cdot 7{y}^{5}}\)
Simplify the radical.	\(\ 6\ \sqrt[4]{16{y}^{4}}\cdot \sqrt[4]{35{y}^{}}\)
Simplify.	\(\ 6\cdot 2y\ \sqrt[4]{35{y}^{}}\)
Multiply.	\(\ 12y\ \sqrt[4]{35{y}^{}}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Use Polynomial Multiplication to Multiply Radical Expressions

In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.

Example

Try it.

Simplify: ⓐ \(\sqrt{6\ }(\sqrt{2}+\sqrt{18})\) ⓑ \(\sqrt[3]{9}\ (5-\sqrt[3]{18}).\)

Solution

ⓐ

	\(\ \sqrt{6}\ (\sqrt{2}+\sqrt{18})\)
Multiply.	\(\ \sqrt{12}+\sqrt{108}\)
Simplify.	\(\ \sqrt{4}\cdot \sqrt{3}+\sqrt{36}\cdot \sqrt{3}\)
Simplify.	\(\ 2\sqrt{3}+6\sqrt{3}\)
Combine like radicals.	\(\ 8\sqrt{3}\)

ⓑ

	\(\ \sqrt[3]{9}\ (5-\sqrt[3]{18})\)
Distribute.	\(\ 5\sqrt[3]{9}-\sqrt[3]{162}\)
Simplify.	\(\ 5\ \sqrt[3]{9}-\sqrt[3]{27}\cdot \sqrt[3]{6}\)
Simplify.	\(\ 5\ \sqrt[3]{9}-3\ \sqrt[3]{6}\)

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

Example

Try it.

Simplify: ⓐ \((3-2\sqrt{7})(4-2\sqrt{7})\) ⓑ \((\sqrt[3]{x}-2)(\sqrt[3]{x}+4).\)

Solution

ⓐ

	\(\ (3-2\sqrt{7})(4-2\sqrt{7})\)
Multiply.	\(\ 12-6\sqrt{7}-8\sqrt{7}+4\cdot 7\)
Simplify.	\(\ 12-6\sqrt{7}-8\sqrt{7}+28\)
Combine like terms.	\(\ 40-14\sqrt{7}\)

ⓑ

	\(\ (\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)
Multiply.	\(\ \sqrt[3]{{x}^{2}}+4\ \sqrt[3]{x}-2\ \sqrt[3]{x}-8\)
Combine like terms.	\(\ \sqrt[3]{{x}^{2}}+2\ \sqrt[3]{x}-8\)

Example

Try it.

Simplify: \((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5}).\)

Solution

	\(\ (3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
Multiply.	\(\ 3\cdot 2+12\sqrt{10}-\sqrt{10}-4\cdot 5\)
Simplify.	\(\ 6+12\sqrt{10}-\sqrt{10}-20\)
Combine like terms.	\(\ -14+11\sqrt{10}\)

Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.

We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

Product Property of Roots
- For any real numbers, \(\sqrt[n]{a}\) and \(\sqrt[n]{b},\) and for any integer \(n\ge 2\)
  \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}\) and \(\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}\)
Special Products
\(\begin{array}{llllll}\text{Binomial Squares} & & & & & \text{Product of Conjugates} \\ {(a+b)}^{2}={a}^{2}+2ab+{b}^{2} & & & & & (a+b)(a-b)={a}^{2}-{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} & & & & & \end{array}\)

Add, Subtract, and Multiply Radical Expressions

Add and Subtract Radical Expressions

In the following exercises, simplify.

Try it.

ⓐ \(8\sqrt{2}-5\sqrt{2}\) ⓑ \(5\ \sqrt[3]{m}+2\ \sqrt[3]{m}\) ⓒ \(8\ \sqrt[4]{m}-2\ \sqrt[4]{m}\)

Solution

ⓐ \(3\sqrt{2}\) ⓑ \(7\sqrt[3]{m}\) ⓒ \(6\sqrt[4]{m}\)

Try it.

ⓐ \(7\sqrt{2}-3\sqrt{2}\) ⓑ \(7\ \sqrt[3]{p}+2\ \sqrt[3]{p}\) ⓒ \(5\ \sqrt[3]{x}-3\ \sqrt[3]{x}\)

Try it.

ⓐ \(3\sqrt{5}+6\sqrt{5}\) ⓑ \(9\ \sqrt[3]{a}+3\ \sqrt[3]{a}\) ⓒ \(5\ \sqrt[4]{2z}+\sqrt[4]{2z}\)

Solution

ⓐ \(9\sqrt{5}\) ⓑ \(12\sqrt[3]{a}\) ⓒ \(6\sqrt[4]{2z}\)

Try it.

ⓐ \(4\sqrt{5}+8\sqrt{5}\) ⓑ \(\sqrt[3]{m}-4\ \sqrt[3]{m}\) ⓒ \(\sqrt{n}+3\sqrt{n}\)

Try it.

ⓐ \(3\sqrt{2a}-4\sqrt{2a}+5\sqrt{2a}\) ⓑ \(5\ \sqrt[4]{3ab}-3\ \sqrt[4]{3ab}-2\ \sqrt[4]{3ab}\)

Solution

ⓐ \(4\sqrt{2a}\) ⓑ 0

Try it.

ⓐ \(\sqrt{11b}-5\sqrt{11b}+3\sqrt{11b}\) ⓑ \(8\ \sqrt[4]{11cd}+5\ \sqrt[4]{11cd}-9\ \sqrt[4]{11cd}\)

Try it.

ⓐ \(8\sqrt{3c}+2\sqrt{3c}-9\sqrt{3c}\) ⓑ \(2\ \sqrt[3]{4pq}-5\ \sqrt[3]{4pq}+4\ \sqrt[3]{4pq}\)

Solution

ⓐ \(\sqrt{3c}\) ⓑ \(\sqrt[3]{4pq}\)

Try it.

ⓐ \(3\sqrt{5d}+8\sqrt{5d}-11\sqrt{5d}\) ⓑ \(11\ \sqrt[3]{2rs}-9\ \sqrt[3]{2rs}+3\ \sqrt[3]{2rs}\)

Try it.

ⓐ \(\sqrt{27}-\sqrt{75}\) ⓑ \(\sqrt[3]{40}-\sqrt[3]{320}\) ⓒ \(\frac{1}{2}\ \sqrt[4]{32}+\frac{2}{3}\ \sqrt[4]{162}\)

Solution

ⓐ \(-2\sqrt{3}\) ⓑ \(-2\sqrt[3]{5}\) ⓒ \(3\sqrt[4]{2}\)

Try it.

ⓐ \(\sqrt{72}-\sqrt{98}\) ⓑ \(\sqrt[3]{24}+\sqrt[3]{81}\) ⓒ \(\frac{1}{2}\ \sqrt[4]{80}-\frac{2}{3}\ \sqrt[4]{405}\)

Try it.

ⓐ \(\sqrt{48}+\sqrt{27}\) ⓑ \(\sqrt[3]{54}+\sqrt[3]{128}\) ⓒ \(6\ \sqrt[4]{5}-\frac{3}{2}\ \sqrt[4]{80}\)

Solution

ⓐ \(7\sqrt{3}\) ⓑ \(7\sqrt[3]{2}\) ⓒ \(3\sqrt[4]{5}\)

Try it.

ⓐ \(\sqrt{45}+\sqrt{80}\) ⓑ \(\sqrt[3]{81}-\sqrt[3]{192}\) ⓒ \(\frac{5}{2}\ \sqrt[4]{80}+\frac{7}{3}\ \sqrt[4]{405}\)

Try it.

ⓐ \(\sqrt{72{a}^{5}}-\sqrt{50{a}^{5}}\) ⓑ \(9\ \sqrt[4]{80{p}^{4}}-6\ \sqrt[4]{405{p}^{4}}\)

Solution

ⓐ \({a}^{2}\sqrt{2a}\) ⓑ 0

Try it.

ⓐ \(\sqrt{48{b}^{5}}-\sqrt{75{b}^{5}}\) ⓑ \(8\ \sqrt[3]{64{q}^{6}}-3\ \sqrt[3]{125{q}^{6}}\)

Try it.

ⓐ \(\sqrt{80{c}^{7}}-\sqrt{20{c}^{7}}\) ⓑ \(2\ \sqrt[4]{162{r}^{10}}+4\ \sqrt[4]{32{r}^{10}}\)

Solution

ⓐ \(2{c}^{3}\sqrt{5c}\) ⓑ \(14{r}^{2}\sqrt[4]{2{r}^{2}}\)

Try it.

ⓐ \(\sqrt{96{d}^{9}}-\sqrt{24{d}^{9}}\) ⓑ \(5\ \sqrt[4]{243{s}^{6}}+2\ \sqrt[4]{3{s}^{6}}\)

Try it.

\(3\ \sqrt{128{y}^{2}}+4y\ \sqrt{162}-8\ \sqrt{98{y}^{2}}\)

Solution

\(4y\sqrt{2}\)

Try it.

\(3\ \sqrt{75{y}^{2}}+8y\ \sqrt{48}-\sqrt{300{y}^{2}}\)

Multiply Radical Expressions

In the following exercises, simplify.

Try it.

ⓐ \((-2\sqrt[]{3})(3\sqrt[]{18})\) ⓑ \((8\ \sqrt[3]{4})(-4\ \sqrt[3]{18})\)

Solution

ⓐ \(-18\sqrt{6}\) ⓑ \(-64\sqrt[3]{9}\)

Try it.

ⓐ \((-4\sqrt[]{5})(5\sqrt[]{10})\) ⓑ \((-2\ \sqrt[3]{9})(7\ \sqrt[3]{9})\)

Try it.

ⓐ \((5\sqrt[]{6})(\text{-}\sqrt[]{12})\) ⓑ \((-2\ \sqrt[4]{18})(\text{-}\ \sqrt[4]{9})\)

Solution

ⓐ \(-30\sqrt{2}\) ⓑ \(6\sqrt[4]{2}\)

Try it.

ⓐ \((-2\sqrt{7})(-2\sqrt{14})\) ⓑ \((-3\ \sqrt[4]{8})(-5\ \sqrt[4]{6})\)

Try it.

ⓐ \((4\sqrt{12{z}^{3}})(3\sqrt{9z})\) ⓑ \((5\ \sqrt[3]{3{x}^{3}})(3\ \sqrt[3]{18{x}^{3}})\)

Solution

ⓐ \(72{z}^{2}\sqrt{3}\) ⓑ \(45{x}^{2}\sqrt[3]{2}\)

Try it.

ⓐ \((3\sqrt{2{x}^{3}})(7\sqrt{18{x}^{2}})\) ⓑ \((-6\ \sqrt[3]{20{a}^{2}})(-2\ \sqrt[3]{16{a}^{3}})\)

Try it.

ⓐ \((-2\sqrt{7{z}^{3}})(3\sqrt{14{z}^{8}})\) ⓑ \((2\ \sqrt[4]{8{y}^{2}})(-2\ \sqrt[4]{12{y}^{3}})\)

Solution

ⓐ \(-42{z}^{5}\sqrt{2z}\) ⓑ \(-8y\sqrt[4]{6y}\)

Try it.

ⓐ \((4\sqrt{2{k}^{5}})(-3\sqrt{32{k}^{6}})\) ⓑ \((\text{-}\ \sqrt[4]{6{b}^{3}})(3\ \sqrt[4]{8{b}^{3}})\)

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

Try it.

ⓐ \(\sqrt{7}(5+2\sqrt{7})\) ⓑ \(\sqrt[3]{6}\ (4+\sqrt[3]{18})\)

Solution

ⓐ \(14+5\sqrt{7}\) ⓑ \(4\sqrt[3]{6}+3\sqrt[3]{4}\)

Try it.

ⓐ \(\sqrt{11}(8+4\sqrt{11})\) ⓑ \(\sqrt[3]{3}\ (\sqrt[3]{9}+\sqrt[3]{18})\)

Try it.

ⓐ \(\sqrt{11}(-3+4\sqrt{11})\) ⓑ \(\sqrt[4]{3}\ (\sqrt[4]{54}+\sqrt[4]{18})\)

Solution

ⓐ \(44-3\sqrt{11}\) ⓑ \(3\sqrt[4]{2}+\sqrt[4]{54}\)

Try it.

ⓐ \(\sqrt{2}(-5+9\sqrt{2})\) ⓑ \(\sqrt[4]{2}\ (\sqrt[4]{12}+\sqrt[4]{24})\)

Try it.

\((7+\sqrt{3})(9-\sqrt{3})\)

Solution

\(60+2\sqrt{3}\)

Try it.

\((8-\sqrt{2})(3+\sqrt{2})\)

Try it.

ⓐ \((9-3\sqrt{2})(6+4\sqrt{2})\) ⓑ \((\sqrt[3]{x}-3)(\sqrt[3]{x}+1)\)

Solution

ⓐ \(30+18\sqrt{2}\) ⓑ \(\sqrt[3]{{x}^{2}}-2\sqrt[3]{x}-3\)

Try it.

ⓐ \((3-2\sqrt{7})(5-4\sqrt{7})\) ⓑ \((\sqrt[3]{x}-5)(\sqrt[3]{x}-3)\)

Try it.

ⓐ \((1+3\sqrt{10})(5-2\sqrt{10})\) ⓑ \((2\ \sqrt[3]{x}+6)(\sqrt[3]{x}+1)\)

Solution

ⓐ \(-55+13\sqrt{10}\)
ⓑ \(2\sqrt[3]{{x}^{2}}+8\sqrt[3]{x}+6\)

Try it.

ⓐ \((7-2\sqrt{5})(4+9\sqrt{5})\) ⓑ \((3\ \sqrt[3]{x}+2)(\sqrt[3]{x}-2)\)

Try it.

\((\sqrt{3}+\sqrt{10})(\sqrt{3}+2\sqrt{10})\)

Solution

\(23+3\sqrt{30}\)

Try it.

\((\sqrt{11}+\sqrt{5})(\sqrt{11}+6\sqrt{5})\)

Try it.

\((2\sqrt{7}-5\sqrt{11})(4\sqrt{7}+9\sqrt{11})\)

Solution

\(-439-2\sqrt{77}\)

Try it.

\((4\sqrt{6}+7\sqrt{13})(8\sqrt{6}-3\sqrt{13})\)

Try it.

ⓐ \({(3+\sqrt{5})}^{2}\) ⓑ \({(2-5\sqrt{3})}^{2}\)

Solution

ⓐ \(14+6\sqrt{5}\) ⓑ \(79-20\sqrt{3}\)

Try it.

ⓐ \({(4+\sqrt{11})}^{2}\) ⓑ \({(3-2\sqrt{5})}^{2}\)

Try it.

ⓐ \({(9-\sqrt{6})}^{2}\) ⓑ \({(10+3\sqrt{7})}^{2}\)

Solution

ⓐ \(87-18\sqrt{6}\)
ⓑ \(163+60\sqrt{7}\)

Try it.

ⓐ \({(5-\sqrt{10})}^{2}\) ⓑ \({(8+3\sqrt{2})}^{2}\)

Try it.

\((4+\sqrt{2})(4-\sqrt{2})\)

Solution

Try it.

\((7+\sqrt{10})(7-\sqrt{10})\)

Try it.

\((4+9\sqrt{3})(4-9\sqrt{3})\)

Solution

\(-227\)

Try it.

\((1+8\sqrt{2})(1-8\sqrt{2})\)

Try it.

\((12-5\sqrt{5})(12+5\sqrt{5})\)

Solution

\(19\)

Try it.

\((9-4\sqrt{3})(9+4\sqrt{3})\)

Try it.

\((\sqrt[3]{3x}+2)(\sqrt[3]{3x}-2)\)

Solution

\(\sqrt[3]{9{x}^{2}}-4\)

Try it.

\((\sqrt[3]{4x}+3)(\sqrt[3]{4x}-3)\)

Mixed Practice

Try it.

\(\frac{2}{3}\sqrt{27}+\frac{3}{4}\sqrt{48}\)

Solution

\(5\sqrt{3}\)

Try it.

\(\sqrt{175{k}^{4}}-\sqrt{63{k}^{4}}\)

Try it.

\(\frac{5}{6}\sqrt{162}+\frac{3}{16}\sqrt{128}\)

Solution

\(9\sqrt{2}\)

Try it.

\(\sqrt[3]{24}+\sqrt[3]{\text{/}81}\)

Try it.

\(\frac{1}{2}\ \sqrt[4]{80}-\frac{2}{3}\ \sqrt[4]{405}\)

Solution

\(\text{-}\sqrt[4]{5}\)

Try it.

\(8\sqrt[4]{13}-4\sqrt[4]{13}-3\sqrt[4]{13}\)

Try it.

\(5\sqrt{12{c}^{4}}-3\sqrt{27{c}^{6}}\)

Solution

\(10{c}^{2}\sqrt{3}-9{c}^{3}\sqrt{3}\)

Try it.

\(\sqrt{80{a}^{5}}-\sqrt{45{a}^{5}}\)

Try it.

\(\frac{3}{5}\sqrt{75}-\frac{1}{4}\sqrt{48}\)

Solution

\(2\sqrt{3}\)

Try it.

\(21\ \sqrt[3]{9}-2\ \sqrt[3]{9}\)

Try it.

\(8\ \sqrt[3]{64{q}^{6}}-3\ \sqrt[3]{125{q}^{6}}\)

Solution

\(17{q}^{2}\)

Try it.

\(11\sqrt{11}-10\sqrt{11}\)

Try it.

\(\sqrt{3}\cdot \sqrt{21}\)

Solution

\(3\sqrt{7}\)

Try it.

\((4\sqrt{6})(\text{-}\sqrt{18})\)

Try it.

\((7\sqrt[3]{4})(-3\sqrt[3]{18})\)

Solution

\(-42\sqrt[3]{9}\)

Try it.

\((4\sqrt{12{x}^{5}})(2\sqrt{6{x}^{3}})\)

Try it.

\({(\sqrt{29})}^{2}\)

Solution

Try it.

\((-4\sqrt{17})(-3\sqrt{17})\)

Try it.

\((-4+\sqrt{17})(-3+\sqrt{17})\)

Solution

\(29-7\sqrt{17}\)

Try it.

\((3\ \sqrt[4]{8{a}^{2}})(\sqrt[4]{12{a}^{3}})\)

Try it.

\({(6-3\sqrt{2})}^{2}\)

Solution

\(54-36\sqrt{2}\)

Try it.

\(\sqrt{3}(4-3\sqrt{3})\)

Try it.

\(\sqrt[3]{3}\ (2\ \sqrt[3]{9}+\sqrt[3]{18})\)

Solution

\(6+3\sqrt[3]{2}\)

Try it.

\((\sqrt{6}+\sqrt{3})(\sqrt{6}+6\sqrt{3})\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Add: \(3{x}^{2}+9x-5-({x}^{2}-2x+3).\)
If you missed this problem, review .

Αποκάλυψέ την.

\(2{x}^{2}+11x-8\)
Simplify: \((2+a)(4-a).\)
If you missed this problem, review .

Αποκάλυψέ την.

\(8+2a-{a}^{2}\)
Simplify: \({(9-5y)}^{2}.\)
If you missed this problem, review .

Αποκάλυψέ την.

\(81-90y+25{y}^{2}\)
Simplify: ⓐ \(2\ \sqrt{2}-7\ \sqrt{2}\) ⓑ \(5\ \sqrt[3]{y}+4\ \sqrt[3]{y}\) ⓒ \(7\ \sqrt[4]{x}-2\ \sqrt[4]{y}.\)

Αποκάλυψέ την.

ⓐ

\(\ 2\ \sqrt{2}-7\ \sqrt{2}\)
Since the radicals are like, we subtract the
coefficients. \(\ -5\ \sqrt{2}\)
ⓑ

\(\ 5\ \sqrt[3]{y}+4\ \sqrt[3]{y}\)
Since the radicals are like, we add the
coefficients. \(\ 9\ \sqrt[3]{y}\)
ⓒ

\(\ 7\ \sqrt[4]{x}-2\ \sqrt[4]{y}\)
The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.
Simplify: ⓐ \(8\sqrt{2}-9\sqrt{2}\) ⓑ \(4\sqrt[3]{x}+7\sqrt[3]{x}\) ⓒ \(3\sqrt[4]{x}-5\sqrt[4]{y}.\)

Αποκάλυψέ την.

ⓐ \(\text{-}\sqrt{2}\) ⓑ \(11\sqrt[3]{x}\)
ⓒ \(3\sqrt[4]{x}-5\sqrt[4]{y}\)
Simplify: ⓐ \(5\sqrt{3}-9\sqrt{3}\) ⓑ \(5\sqrt[3]{y}+3\sqrt[3]{y}\) ⓒ \(5\sqrt[4]{m}-2\sqrt[3]{m}.\)

Αποκάλυψέ την.

ⓐ \(-4\sqrt{3}\) ⓑ \(8\sqrt[3]{y}\)
ⓒ \(5\sqrt[4]{m}-2\sqrt[3]{m}\)
Simplify: ⓐ \(2\ \sqrt{5n}-6\ \sqrt{5n}+4\ \sqrt{5n}\) ⓑ \(\sqrt[4]{3xy}+5\ \sqrt[4]{3xy}-4\ \sqrt[4]{3xy}.\)

Αποκάλυψέ την.

ⓐ

\(\ 2\ \sqrt{5n}-6\ \sqrt{5n}+4\ \sqrt{5n}\)
Since the radicals are like, we combine them. \(\ 0\ \sqrt{5n}\)
Simplify. \(\ 0\)
ⓑ

\(\ \sqrt[4]{3xy}+5\ \sqrt[4]{3xy}-4\ \sqrt[4]{3xy}\)
Since the radicals are like, we combine them. \(\ 2\ \sqrt[4]{3xy}\)
Simplify: ⓐ \(\sqrt{7x}-7\ \sqrt{7x}+4\ \sqrt{7x}\) ⓑ \(4\ \sqrt[4]{5xy}+2\ \sqrt[4]{5xy}-7\ \sqrt[4]{5xy}.\)

Αποκάλυψέ την.

ⓐ \(-2\sqrt{7x}\) ⓑ \(\text{-}\sqrt[4]{5xy}\)
Simplify: ⓐ \(4\ \sqrt{3y}-7\ \sqrt{3y}+2\ \sqrt{3y}\) ⓑ \(6\ \sqrt[3]{7mn}+\sqrt[3]{7mn}-4\ \sqrt[3]{7mn}.\)

Αποκάλυψέ την.

ⓐ \(\text{-}\sqrt{3y}\) ⓑ \(3\sqrt[3]{7mn}\)

Simplify: ⓐ \(\sqrt{20}+3\sqrt{5}\) ⓑ \(\sqrt[3]{24}-\sqrt[3]{375}\) ⓒ \(\frac{1}{2}\sqrt[4]{48}-\frac{2}{3}\sqrt[4]{243}.\)

Αποκάλυψέ την.

ⓐ

	\(\ \sqrt{20}+3\ \sqrt{5}\)
Simplify the radicals, when possible.	\(\ \sqrt{4}\cdot \sqrt{5}+3\ \sqrt{5}\)
	\(\ 2\ \sqrt{5}+3\ \sqrt{5}\)
Combine the like radicals.	\(\ 5\ \sqrt{5}\)

ⓑ

	\(\ \sqrt[3]{24}-\sqrt[3]{375}\)
Simplify the radicals.	\(\ \sqrt[3]{8}\cdot \sqrt[3]{3}-\sqrt[3]{125}\cdot \sqrt[3]{3}\)
	\(\ 2\ \sqrt[3]{3}-5\ \sqrt[3]{3}\)
Combine the like radicals.	\(\ -3\sqrt[3]{3}\)

ⓒ

	\(\ \frac{1}{2}\ \sqrt[4]{48}-\frac{2}{3}\ \sqrt[4]{243}\)
Simplify the radicals.	\(\ \frac{1}{2}\ \sqrt[4]{16}\cdot \sqrt[4]{3}-\frac{2}{3}\ \sqrt[4]{81}\cdot \sqrt[4]{3}\)
	\(\ \frac{1}{2}\cdot 2\cdot \sqrt[4]{3}-\frac{2}{3}\cdot 3\cdot \sqrt[4]{3}\)
	\(\ \sqrt[4]{3}-2\ \sqrt[4]{3}\)
Combine the like radicals.	\(\ \text{-}\sqrt[4]{3}\)

Simplify: ⓐ \(\sqrt{18}+6\ \sqrt{2}\) ⓑ \(6\ \sqrt[3]{16}-2\ \sqrt[3]{250}\) ⓒ \(\frac{2}{3}\ \sqrt[3]{81}-\frac{1}{2}\ \sqrt[3]{24}.\)

Αποκάλυψέ την.

ⓐ \(9\sqrt{2}\) ⓑ \(2\sqrt[3]{2}\) ⓒ \(\sqrt[3]{3}\)
Simplify: ⓐ \(\sqrt{27}+4\ \sqrt{3}\) ⓑ \(4\ \sqrt[3]{5}-7\ \sqrt[3]{40}\) ⓒ \(\frac{1}{2}\ \sqrt[3]{128}-\frac{5}{3}\ \sqrt[3]{54}.\)

Αποκάλυψέ την.

ⓐ \(7\sqrt{3}\) ⓑ \(-10\sqrt[3]{5}\) ⓒ \(-3\sqrt[3]{2}\)

Simplify: ⓐ \(9\sqrt{50{m}^{2}}-6\sqrt{48{m}^{2}}\) ⓑ \(\sqrt[3]{54{n}^{5}}-\sqrt[3]{16{n}^{5}}.\)

Αποκάλυψέ την.

ⓐ

	\(\ 9\ \sqrt{50{m}^{2}}-6\ \sqrt{48{m}^{2}}\)
Simplify the radicals.	\(\ 9\ \sqrt{25{m}^{2}}\cdot \sqrt{2}-6\ \sqrt{16{m}^{2}}\cdot \sqrt{3}\)
	\(\ 9\cdot 5m\cdot \sqrt{2}-6\cdot 4m\cdot \sqrt{3}\)
	\(\ 45m\ \sqrt{2}-24m\ \sqrt{3}\)
The radicals are not like and so cannot be combined.

ⓑ

	\(\ \sqrt[3]{54{n}^{5}}-\sqrt[3]{16{n}^{5}}\)
Simplify the radicals.	\(\ \sqrt[3]{27{n}^{3}}\cdot \sqrt[3]{2{n}^{2}}-\sqrt[3]{8{n}^{3}}\cdot \sqrt[3]{2{n}^{2}}\)
	\(\ 3n\ \sqrt[3]{2{n}^{2}}-2n\ \sqrt[3]{2{n}^{2}}\)
Combine the like radicals.	\(\ n\ \sqrt[3]{2{n}^{2}}\)

Simplify: ⓐ \(\sqrt{32{m}^{7}}-\sqrt{50{m}^{7}}\) ⓑ \(\sqrt[3]{135{x}^{7}}-\sqrt[3]{40{x}^{7}}.\)

Αποκάλυψέ την.

ⓐ \(\text{-}{m}^{3}\sqrt{2m}\) ⓑ \({x}^{2}\sqrt[3]{5x}\)
Simplify: ⓐ \(\sqrt{27{p}^{3}}-\sqrt{48{p}^{3}}\) ⓑ \(\sqrt[3]{256{y}^{5}}-\sqrt[3]{32{y}^{5}}.\)

Αποκάλυψέ την.

ⓐ \(\text{-}p\sqrt{3p}\)
ⓑ \(4y\sqrt[3]{4{y}^{2}}-2y\sqrt[3]{4{n}^{2}}\)

Simplify: ⓐ \((6\ \sqrt{2})(3\ \sqrt{10})\) ⓑ \((-5\ \sqrt[3]{4})(-4\ \sqrt[3]{6}).\)

Αποκάλυψέ την.

ⓐ

	\(\ (6\ \sqrt{2})(3\ \sqrt{10})\)
Multiply using the Product Property.	\(\ 18\ \sqrt{20}\)
Simplify the radical.	\(\ 18\ \sqrt{4}\cdot \sqrt{5}\)
Simplify.	\(\ 18\cdot 2\cdot \sqrt{5}\)
	\(\ 36\ \sqrt{5}\)

ⓑ

	\(\ (-5\ \sqrt[3]{4})(-4\ \sqrt[3]{6})\)
Multiply using the Product Property.	\(\ 20\ \sqrt[3]{24}\)
Simplify the radical.	\(\ 20\ \sqrt[3]{8}\cdot \sqrt[3]{3}\)
Simplify.	\(\ 20\cdot 2\cdot \sqrt[3]{3}\)
	\(\ 40\ \sqrt[3]{3}\)

Simplify: ⓐ \((3\sqrt{2})(2\sqrt{30})\) ⓑ \((2\ \sqrt[3]{18})(-3\ \sqrt[3]{6}).\)

Αποκάλυψέ την.

ⓐ \(12\sqrt{15}\) ⓑ \(-18\sqrt[3]{4}\)
Simplify: ⓐ \((3\sqrt{3})(3\sqrt{6})\) ⓑ \((-4\ \sqrt[3]{9})(3\ \sqrt[3]{6}).\)

Αποκάλυψέ την.

ⓐ \(27\sqrt{2}\) ⓑ \(-36\sqrt[3]{2}\)

Simplify: ⓐ \((10\ \sqrt{6{p}^{3}})(4\ \sqrt{3p})\) ⓑ \((2\ \sqrt[4]{20{y}^{2}})(3\ \sqrt[4]{28{y}^{3}}).\)

Αποκάλυψέ την.

ⓐ

	\(\ (10\sqrt{6{p}^{3}})(4\sqrt{3p})\)
Multiply.	\(\ 40\sqrt{18{p}^{4}}\)
Simplify the radical.	\(\ 40\sqrt{9{p}^{4}}\cdot \sqrt{2}\)
Simplify.	\(\ 40\cdot 3{p}^{2}\cdot \sqrt{2}\)
	\(\ 120{p}^{2}\sqrt{2}\)

ⓑ When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.

	\(\ (2\sqrt[4]{20{y}^{2}})(3\sqrt[4]{28{y}^{3}})\)
Multiply.	\(\ 6\ \sqrt[4]{4\cdot 5\cdot 4\cdot 7{y}^{5}}\)
Simplify the radical.	\(\ 6\ \sqrt[4]{16{y}^{4}}\cdot \sqrt[4]{35{y}^{}}\)
Simplify.	\(\ 6\cdot 2y\ \sqrt[4]{35{y}^{}}\)
Multiply.	\(\ 12y\ \sqrt[4]{35{y}^{}}\)

Simplify: ⓐ \((6\sqrt{6{x}^{2}})(8\sqrt{30{x}^{4}})\) ⓑ \((-4\ \sqrt[4]{12{y}^{3}})(\text{-}\sqrt[4]{8{y}^{3}}).\)

Αποκάλυψέ την.

ⓐ \(288{x}^{3}\sqrt{5}\) ⓑ \(8y\sqrt[4]{6{y}^{2}}\)
Simplify: ⓐ \((2\sqrt{6{y}^{4}})(12\sqrt{30y})\) ⓑ \((-4\ \sqrt[4]{9{a}^{3}})(3\ \sqrt[4]{27{a}^{2}}).\)

Αποκάλυψέ την.

ⓐ \(144{y}^{2}\sqrt{5y}\) ⓑ \(-36a\sqrt[4]{3a}\)

Simplify: ⓐ \(\sqrt{6\ }(\sqrt{2}+\sqrt{18})\) ⓑ \(\sqrt[3]{9}\ (5-\sqrt[3]{18}).\)

Αποκάλυψέ την.

ⓐ

	\(\ \sqrt{6}\ (\sqrt{2}+\sqrt{18})\)
Multiply.	\(\ \sqrt{12}+\sqrt{108}\)
Simplify.	\(\ \sqrt{4}\cdot \sqrt{3}+\sqrt{36}\cdot \sqrt{3}\)
Simplify.	\(\ 2\sqrt{3}+6\sqrt{3}\)
Combine like radicals.	\(\ 8\sqrt{3}\)

ⓑ

	\(\ \sqrt[3]{9}\ (5-\sqrt[3]{18})\)
Distribute.	\(\ 5\sqrt[3]{9}-\sqrt[3]{162}\)
Simplify.	\(\ 5\ \sqrt[3]{9}-\sqrt[3]{27}\cdot \sqrt[3]{6}\)
Simplify.	\(\ 5\ \sqrt[3]{9}-3\ \sqrt[3]{6}\)

Simplify: ⓐ \(\sqrt{6}(1+3\sqrt{6})\) ⓑ \(\sqrt[3]{4}(-2-\sqrt[3]{6}).\)

Αποκάλυψέ την.

ⓐ \(18+\sqrt{6}\) ⓑ \(-2\sqrt[3]{4}-2\sqrt[3]{3}\)
Simplify: ⓐ \(\sqrt{8}(2-5\sqrt{8})\) ⓑ \(\sqrt[3]{3}(\text{-}\sqrt[3]{9}-\sqrt[3]{6}).\)

Αποκάλυψέ την.

ⓐ \(-40+4\sqrt{2}\) ⓑ \(-3-\sqrt[3]{18}\)
Simplify: ⓐ \((3-2\sqrt{7})(4-2\sqrt{7})\) ⓑ \((\sqrt[3]{x}-2)(\sqrt[3]{x}+4).\)

Αποκάλυψέ την.

ⓐ

\(\ (3-2\sqrt{7})(4-2\sqrt{7})\)
Multiply. \(\ 12-6\sqrt{7}-8\sqrt{7}+4\cdot 7\)
Simplify. \(\ 12-6\sqrt{7}-8\sqrt{7}+28\)
Combine like terms. \(\ 40-14\sqrt{7}\)
ⓑ

\(\ (\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)
Multiply. \(\ \sqrt[3]{{x}^{2}}+4\ \sqrt[3]{x}-2\ \sqrt[3]{x}-8\)
Combine like terms. \(\ \sqrt[3]{{x}^{2}}+2\ \sqrt[3]{x}-8\)
Simplify: ⓐ \((6-3\sqrt{7})(3+4\sqrt{7})\) ⓑ \((\sqrt[3]{x}-2)(\sqrt[3]{x}-3).\)

Αποκάλυψέ την.

ⓐ \(-66+15\sqrt{7}\)
ⓑ \(\sqrt[3]{{x}^{2}}-5\sqrt[3]{x}+6\)
Simplify: ⓐ \((2-3\sqrt{11})(4-\sqrt{11})\) ⓑ \((\sqrt[3]{x}+1)(\sqrt[3]{x}+3).\)

Αποκάλυψέ την.

ⓐ \(41-14\sqrt{11}\)
ⓑ \(\sqrt[3]{{x}^{2}}+4\sqrt[3]{x}+3\)
Simplify: \((3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5}).\)

Αποκάλυψέ την.

\(\ (3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
Multiply. \(\ 3\cdot 2+12\sqrt{10}-\sqrt{10}-4\cdot 5\)
Simplify. \(\ 6+12\sqrt{10}-\sqrt{10}-20\)
Combine like terms. \(\ -14+11\sqrt{10}\)
Simplify: \((5\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})\)

Αποκάλυψέ την.

\(1+9\sqrt{21}\)
Simplify: \((\sqrt{6}-3\sqrt{8})(2\sqrt{6}+\sqrt{8})\)

Αποκάλυψέ την.

\(-12-20\sqrt{3}\)
Simplify: ⓐ \({(2+\sqrt{3})}^{2}\) ⓑ \({(4-2\sqrt{5})}^{2}.\)

Αποκάλυψέ την.

Be sure to include the \(2ab\) term when squaring a binomial.
ⓐ

Multiply, using the Product of Binomial Squares Pattern.
Simplify.
Combine like terms.

ⓑ

Multiply, using the Product of Binomial Squares Pattern.
Simplify.
Combine like terms.
Simplify: ⓐ \({(10+\sqrt{2})}^{2}\) ⓑ \({(1+3\sqrt{6})}^{2}.\)

Αποκάλυψέ την.

ⓐ \(102+20\sqrt{2}\) ⓑ \(55+6\sqrt{6}\)
Simplify: ⓐ \({(6-\sqrt{5})}^{2}\) ⓑ \({(9-2\sqrt{10})}^{2}.\)

Αποκάλυψέ την.

ⓐ \(41-12\sqrt{5}\)
ⓑ \(121-36\sqrt{10}\)
Simplify: \((5-2\sqrt{3})(5+2\sqrt{3}).\)

Αποκάλυψέ την.

Multiply, using the Product of Conjugates Pattern.
Simplify.
Simplify: \((3-2\sqrt{5})(3+2\sqrt{5})\)

Αποκάλυψέ την.

\(-11\)
Simplify: \((4+5\sqrt{7})(4-5\sqrt{7}).\)

Αποκάλυψέ την.

\(-159\)
ⓐ \(8\sqrt{2}-5\sqrt{2}\) ⓑ \(5\ \sqrt[3]{m}+2\ \sqrt[3]{m}\) ⓒ \(8\ \sqrt[4]{m}-2\ \sqrt[4]{m}\)

Αποκάλυψέ την.

ⓐ \(3\sqrt{2}\) ⓑ \(7\sqrt[3]{m}\) ⓒ \(6\sqrt[4]{m}\)
ⓐ \(3\sqrt{5}+6\sqrt{5}\) ⓑ \(9\ \sqrt[3]{a}+3\ \sqrt[3]{a}\) ⓒ \(5\ \sqrt[4]{2z}+\sqrt[4]{2z}\)

Αποκάλυψέ την.

ⓐ \(9\sqrt{5}\) ⓑ \(12\sqrt[3]{a}\) ⓒ \(6\sqrt[4]{2z}\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Add, Subtract, and Multiply Radical Expressions

Add and subtract radical expressions
Multiply radical expressions
Use polynomial multiplication to multiply radical expressions
For any real numbers,

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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Multiply, using the Product of Binomial Squares Pattern.
Simplify.
Combine like terms.