maths.free › Algebra › Roots and Radicals › Add and Subtract Square Roots

Add and Subtract Square Roots

Add and subtract like square roots

Add and Subtract Like Square Roots

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

Example

Try it.

Simplify: \(2\sqrt{2}-7\sqrt{2}\).

Solution

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

Example

Try it.

Simplify: \(3\sqrt{y}+4\sqrt{y}\).

Solution

	\(3\sqrt{y}+4\sqrt{y}\)
Since the radicals are like, we add the coefficients.	\(7\sqrt{y}\)

Example

Try it.

Simplify: \(4\sqrt{x}-2\sqrt{y}\).

Solution

	\(4\sqrt{x}-2\sqrt{y}\)
Since the radicals are not like, we cannot subtract them. We leave the expression as is.	\(4\sqrt{x}-2\sqrt{y}\)

Example

Try it.

Simplify: \(5\sqrt{13}+4\sqrt{13}+2\sqrt{13}\).

Solution

	\(5\sqrt{13}+4\sqrt{13}+2\sqrt{13}\)
Since the radicals are like, we add the coefficients.	\(11\sqrt{13}\)

Example

Try it.

Simplify: \(2\sqrt{6}-6\sqrt{6}+3\sqrt{3}\).

Solution

	\(2\sqrt{6}-6\sqrt{6}+3\sqrt{3}\)
Since the first two radicals are like, we subtract their coefficients.	\(-4\sqrt{6}+3\sqrt{3}\)

Example

Try it.

Simplify: \(2\sqrt{5n}-6\sqrt{5n}+4\sqrt{5n}\).

Solution

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

When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

Example

Try it.

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

Solution

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

Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Example

Try it.

Simplify: \(\sqrt{20}+3\sqrt{5}\).

Solution

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

Example

Try it.

Simplify: \(\sqrt{48}-\sqrt{75}\).

Solution

	\(\sqrt{48}-\sqrt{75}\)
Simplify the radicals.	\(\sqrt{16}\cdot \sqrt{3}-\sqrt{25}\cdot \sqrt{3}\)
	\(4\sqrt{3}-5\sqrt{3}\)
Combine the like radicals.	\(\text{-}\sqrt{3}\)

Just like we use the Associative Property of Multiplication to simplify \(5(3x)\) and get \(15x\), we can simplify \(5(3\sqrt{x})\) and get \(15\sqrt{x}\). We will use the Associative Property to do this in the next example.

Example

Try it.

Simplify: \(5\sqrt{18}-2\sqrt{8}\).

Solution

	\(5\sqrt{18}-2\sqrt{8}\)
Simplify the radicals.	\(5\cdot \sqrt{9}\cdot \sqrt{2}-2\cdot \sqrt{4}\cdot \sqrt{2}\)
	\(5\cdot 3\cdot \sqrt{2}-2\cdot 2\cdot \sqrt{2}\)
	\(15\sqrt{2}-4\sqrt{2}\)
Combine the like radicals.	\(11\sqrt{2}\)

Example

Try it.

Simplify: \(\frac{3}{4}\sqrt{192}-\frac{5}{6}\sqrt{108}\).

Solution

	\(\frac{3}{4}\sqrt{192}-\frac{5}{6}\sqrt{108}\)
Simplify the radicals.	\(\frac{3}{4}\sqrt{64}\cdot \sqrt{3}-\frac{5}{6}\sqrt{36}\cdot \sqrt{3}\)
	\(\frac{3}{4}\cdot 8\cdot \sqrt{3}-\frac{5}{6}\cdot 6\cdot \sqrt{3}\)
	\(6\sqrt{3}-5\sqrt{3}\)
Combine the like radicals.	\(\sqrt{3}\)

Example

Try it.

Simplify: \(\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}\).

Solution

	\(\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}\)
Simplify the radicals.	\(\frac{2}{3}\sqrt{16}\cdot \sqrt{3}-\frac{3}{4}\sqrt{4}\cdot \sqrt{3}\)
	\(\frac{2}{3}\cdot 4\cdot \sqrt{3}-\frac{3}{4}\cdot 2\cdot \sqrt{3}\)
	\(\frac{8}{3}\sqrt{3}-\frac{3}{2}\sqrt{3}\)
Find a common denominator to subtract the coefficients of the like radicals.	\(\frac{16}{6}\sqrt{3}-\frac{9}{6}\sqrt{3}\)
Simplify.	\(\frac{7}{6}\sqrt{3}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

Try it.

\(8\sqrt{2}-5\sqrt{2}\)

Solution

\(3\sqrt{2}\)

Try it.

\(7\sqrt{2}-3\sqrt{2}\)

Try it.

\(3\sqrt{5}+6\sqrt{5}\)

Solution

\(9\sqrt{5}\)

Try it.

\(4\sqrt{5}+8\sqrt{5}\)

Try it.

\(9\sqrt{7}-10\sqrt{7}\)

Solution

\(\text{-}\sqrt{7}\)

Try it.

\(11\sqrt{7}-12\sqrt{7}\)

Try it.

\(7\sqrt{y}+2\sqrt{y}\)

Solution

\(9\sqrt{y}\)

Try it.

\(9\sqrt{n}+3\sqrt{n}\)

Try it.

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

Solution

\(-3\sqrt{a}\)

Try it.

\(\sqrt{b}-6\sqrt{b}\)

Try it.

\(5\sqrt{c}+2\sqrt{c}\)

Solution

\(7\sqrt{c}\)

Try it.

\(7\sqrt{d}+2\sqrt{d}\)

Try it.

\(8\sqrt{a}-2\sqrt{b}\)

Solution

\(8\sqrt{a}-2\sqrt{b}\)

Try it.

\(5\sqrt{c}-3\sqrt{d}\)

Try it.

\(5\sqrt{m}+\sqrt{n}\)

Solution

\(5\sqrt{m}+\sqrt{n}\)

Try it.

\(\sqrt{n}+3\sqrt{p}\)

Try it.

\(8\sqrt{7}+2\sqrt{7}+3\sqrt{7}\)

Solution

\(13\sqrt{7}\)

Try it.

\(6\sqrt{5}+3\sqrt{5}+\sqrt{5}\)

Try it.

\(3\sqrt{11}+2\sqrt{11}-8\sqrt{11}\)

Solution

\(-3\sqrt{11}\)

Try it.

\(2\sqrt{15}+5\sqrt{15}-9\sqrt{15}\)

Try it.

\(3\sqrt{3}-8\sqrt{3}+7\sqrt{5}\)

Solution

\(-5\sqrt{3}+7\sqrt{5}\)

Try it.

\(5\sqrt{7}-8\sqrt{7}+6\sqrt{3}\)

Try it.

\(6\sqrt{2}+2\sqrt{2}-3\sqrt{5}\)

Solution

\(8\sqrt{2}-3\sqrt{5}\)

Try it.

\(7\sqrt{5}+\sqrt{5}-8\sqrt{10}\)

Try it.

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

Solution

\(4\sqrt{2a}\)

Try it.

\(\sqrt{11b}-5\sqrt{11b}+3\sqrt{11b}\)

Try it.

\(8\sqrt{3c}+2\sqrt{3c}-9\sqrt{3c}\)

Solution

\(\sqrt{3c}\)

Try it.

\(3\sqrt{5d}+8\sqrt{5d}-11\sqrt{5d}\)

Try it.

\(5\sqrt{3ab}+\sqrt{3ab}-2\sqrt{3ab}\)

Solution

\(4\sqrt{3ab}\)

Try it.

\(8\sqrt{11cd}+5\sqrt{11cd}-9\sqrt{11cd}\)

Try it.

\(2\sqrt{pq}-5\sqrt{pq}+4\sqrt{pq}\)

Solution

\(\sqrt{pq}\)

Try it.

\(11\sqrt{2rs}-9\sqrt{2rs}+3\sqrt{2rs}\)

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Try it.

\(\sqrt{50}+4\sqrt{2}\)

Solution

\(9\sqrt{2}\)

Try it.

\(\sqrt{48}+2\sqrt{3}\)

Try it.

\(\sqrt{80}-3\sqrt{5}\)

Solution

\(\sqrt{5}\)

Try it.

\(\sqrt{28}-4\sqrt{7}\)

Try it.

\(\sqrt{27}-\sqrt{75}\)

Solution

\(-2\sqrt{3}\)

Try it.

\(\sqrt{72}-\sqrt{98}\)

Try it.

\(\sqrt{48}+\sqrt{27}\)

Solution

\(7\sqrt{3}\)

Try it.

\(\sqrt{45}+\sqrt{80}\)

Try it.

\(2\sqrt{50}-3\sqrt{72}\)

Solution

\(-8\sqrt{2}\)

Try it.

\(3\sqrt{98}-\sqrt{128}\)

Try it.

\(2\sqrt{12}+3\sqrt{48}\)

Solution

\(16\sqrt{3}\)

Try it.

\(4\sqrt{75}+2\sqrt{108}\)

Try it.

\(\frac{2}{3}\sqrt{72}+\frac{1}{5}\sqrt{50}\)

Solution

\(5\sqrt{2}\)

Try it.

\(\frac{2}{5}\sqrt{75}+\frac{3}{4}\sqrt{48}\)

Try it.

\(\frac{1}{2}\sqrt{20}-\frac{2}{3}\sqrt{45}\)

Solution

\(\text{-}\sqrt{5}\)

Try it.

\(\frac{2}{3}\sqrt{54}-\frac{3}{4}\sqrt{96}\)

Try it.

\(\frac{1}{6}\sqrt{27}-\frac{3}{8}\sqrt{48}\)

Solution

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

Try it.

\(\frac{1}{8}\sqrt{32}-\frac{1}{10}\sqrt{50}\)

Try it.

\(\frac{1}{4}\sqrt{98}-\frac{1}{3}\sqrt{128}\)

Solution

\(-\frac{11}{12}\sqrt{2}\)

Try it.

\(\frac{1}{3}\sqrt{24}+\frac{1}{4}\sqrt{54}\)

Try it.

\(\sqrt{72{a}^{5}}-\sqrt{50{a}^{5}}\)

Solution

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

Try it.

\(\sqrt{48{b}^{5}}-\sqrt{75{b}^{5}}\)

Try it.

\(\sqrt{80{c}^{7}}-\sqrt{20{c}^{7}}\)

Solution

\(2{c}^{3}\sqrt{5c}\)

Try it.

\(\sqrt{96{d}^{9}}-\sqrt{24{d}^{9}}\)

Try it.

\(9\sqrt{80{p}^{4}}-6\sqrt{98{p}^{4}}\)

Solution

\(36{p}^{2}\sqrt{5}-42{p}^{2}\sqrt{2}\)

Try it.

\(8\sqrt{72{q}^{6}}-3\sqrt{75{q}^{6}}\)

Try it.

\(2\sqrt{50{r}^{8}}+4\sqrt{54{r}^{8}}\)

Solution

\(10{r}^{4}\sqrt{2}+12{r}^{4}\sqrt{6}\)

Try it.

\(5\sqrt{27{s}^{6}}+2\sqrt{20{s}^{6}}\)

Try it.

\(3\sqrt{20{x}^{2}}-4\sqrt{45{x}^{2}}+5x\sqrt{80}\)

Solution

\(14x\sqrt{5}\)

Try it.

\(2\sqrt{28{x}^{2}}-\sqrt{63{x}^{2}}+6x\sqrt{7}\)

Try it.

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

Solution

\(4y\sqrt{2}\)

Try it.

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

Mixed Practice

Try it.

\(2\sqrt{8}+6\sqrt{8}-5\sqrt{8}\)

Solution

\(6\sqrt{2}\)

Try it.

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

Try it.

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

Solution

\(2{k}^{2}\sqrt{7}\)

Try it.

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

Try it.

\(2\sqrt{363}-2\sqrt{300}\)

Solution

\(2\sqrt{3}\)

Try it.

\(\sqrt{150}+4\sqrt{6}\)

Try it.

\(9\sqrt{2}-8\sqrt{2}\)

Solution

\(\sqrt{2}\)

Try it.

\(5\sqrt{x}-8\sqrt{y}\)

Try it.

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

Solution

\(\sqrt{13}\)

Try it.

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

Try it.

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

Solution

\({a}^{2}\sqrt{5a}\)

Try it.

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

Try it.

\(21\sqrt{19}-2\sqrt{19}\)

Solution

\(19\sqrt{19}\)

Try it.

\(\sqrt{500}+\sqrt{405}\)

Try it.

\(\frac{5}{6}\sqrt{27}+\frac{5}{8}\sqrt{48}\)

Solution

\(5\sqrt{3}\)

Try it.

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

Try it.

\(\sqrt{75}-\sqrt{108}\)

Solution

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

Try it.

\(2\sqrt{98}-4\sqrt{72}\)

Try it.

\(4\sqrt{24{x}^{2}}-\sqrt{54{x}^{2}}+3x\sqrt{6}\)

Solution

\(8x\sqrt{6}\)

Try it.

\(8\sqrt{80{y}^{6}}-6\sqrt{48{y}^{6}}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Nígbà yìí Kò ní kọ́ǹpútà alágbèéká kan tí o lè fi àwọn à

Àwọn Ìṣàmúlò-ètò (40)

Wá gbogbo àwọn ní pàtó nínú àwọn àkọ́lé ní ìbẹrẹ. Fi àwọn

Add: ⓐ \(3x+9x\) ⓑ \(5m+5n\).

Fi àwọn àgbèwọlé hàn

ⓐ \(12x\) ⓑ \(5m+5n\)
Simplify: \(\sqrt{50{x}^{3}}\).

Fi àwọn àgbèwọlé hàn

\(5x\sqrt{2x}\)
Simplify: \(2\sqrt{2}-7\sqrt{2}\).

Fi àwọn àgbèwọlé hàn

\(2\sqrt{2}-7\sqrt{2}\)
Since the radicals are like, we subtract the coefficients. \(-5\sqrt{2}\)
Simplify: \(8\sqrt{2}-9\sqrt{2}\).

Fi àwọn àgbèwọlé hàn

\(\text{-}\sqrt{2}\)
Simplify: \(5\sqrt{3}-9\sqrt{3}\).

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

\(3\sqrt{y}+4\sqrt{y}\)
Since the radicals are like, we add the coefficients. \(7\sqrt{y}\)
Simplify: \(2\sqrt{x}+7\sqrt{x}\).

Fi àwọn àgbèwọlé hàn

\(9\sqrt{x}\)
Simplify: \(5\sqrt{u}+3\sqrt{u}\).

Fi àwọn àgbèwọlé hàn

\(8\sqrt{u}\)
Simplify: \(4\sqrt{x}-2\sqrt{y}\).

Fi àwọn àgbèwọlé hàn

\(4\sqrt{x}-2\sqrt{y}\)
Since the radicals are not like, we cannot
subtract them. We leave the expression as is. \(4\sqrt{x}-2\sqrt{y}\)
Simplify: \(7\sqrt{p}-6\sqrt{q}\).

Fi àwọn àgbèwọlé hàn

\(7\sqrt{p}-6\sqrt{q}\)
Simplify: \(6\sqrt{a}-3\sqrt{b}\).

Fi àwọn àgbèwọlé hàn

\(6\sqrt{a}-3\sqrt{b}\)
Simplify: \(5\sqrt{13}+4\sqrt{13}+2\sqrt{13}\).

Fi àwọn àgbèwọlé hàn

\(5\sqrt{13}+4\sqrt{13}+2\sqrt{13}\)
Since the radicals are like, we add the coefficients. \(11\sqrt{13}\)
Simplify: \(4\sqrt{11}+2\sqrt{11}+3\sqrt{11}\).

Fi àwọn àgbèwọlé hàn

\(9\sqrt{11}\)
Simplify: \(6\sqrt{10}+2\sqrt{10}+3\sqrt{10}\).

Fi àwọn àgbèwọlé hàn

\(11\sqrt{10}\)
Simplify: \(2\sqrt{6}-6\sqrt{6}+3\sqrt{3}\).

Fi àwọn àgbèwọlé hàn

\(2\sqrt{6}-6\sqrt{6}+3\sqrt{3}\)
Since the first two radicals are like, we
subtract their coefficients. \(-4\sqrt{6}+3\sqrt{3}\)
Simplify: \(5\sqrt{5}-4\sqrt{5}+2\sqrt{6}\).

Fi àwọn àgbèwọlé hàn

\(\sqrt{5}+2\sqrt{6}\)
Simplify: \(3\sqrt{7}-8\sqrt{7}+2\sqrt{5}\).

Fi àwọn àgbèwọlé hàn

\(-5\sqrt{7}+2\sqrt{5}\)
Simplify: \(2\sqrt{5n}-6\sqrt{5n}+4\sqrt{5n}\).

Fi àwọn àgbèwọlé hàn

\(2\sqrt{5n}-6\sqrt{5n}+4\sqrt{5n}\)
Since the radicals are like, we combine them. \(0\sqrt{5n}\)
Simplify. 0
Simplify: \(\sqrt{7x}-7\sqrt{7x}+4\sqrt{7x}\).

Fi àwọn àgbèwọlé hàn

\(-2\sqrt{7x}\)
Simplify: \(4\sqrt{3y}-7\sqrt{3y}+2\sqrt{3y}\).

Fi àwọn àgbèwọlé hàn

\(\text{-}\sqrt{3y}\)
Simplify: \(\sqrt{3xy}+5\sqrt{3xy}-4\sqrt{3xy}\).

Fi àwọn àgbèwọlé hàn

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

Fi àwọn àgbèwọlé hàn

\(-2\sqrt{5xy}\)
Simplify: \(3\sqrt{7mn}+\sqrt{7mn}-4\sqrt{7mn}\).

Fi àwọn àgbèwọlé hàn

\(0\)
Simplify: \(\sqrt{20}+3\sqrt{5}\).

Fi àwọn àgbèwọlé hàn

\(\sqrt{20}+3\sqrt{5}\)
Simplify the radicals, when possible. \(\sqrt{4}\cdot \sqrt{5}+3\sqrt{5}\)
\(2\sqrt{5}+3\sqrt{5}\)
Combine the like radicals. \(5\sqrt{5}\)
Simplify: \(\sqrt{18}+6\sqrt{2}\).

Fi àwọn àgbèwọlé hàn

\(9\sqrt{2}\)
Simplify: \(\sqrt{27}+4\sqrt{3}\).

Fi àwọn àgbèwọlé hàn

\(7\sqrt{3}\)
Simplify: \(\sqrt{48}-\sqrt{75}\).

Fi àwọn àgbèwọlé hàn

\(\sqrt{48}-\sqrt{75}\)
Simplify the radicals. \(\sqrt{16}\cdot \sqrt{3}-\sqrt{25}\cdot \sqrt{3}\)
\(4\sqrt{3}-5\sqrt{3}\)
Combine the like radicals. \(\text{-}\sqrt{3}\)
Simplify: \(\sqrt{32}-\sqrt{18}\).

Fi àwọn àgbèwọlé hàn

\(\sqrt{2}\)
Simplify: \(\sqrt{20}-\sqrt{45}\).

Fi àwọn àgbèwọlé hàn

\(\text{-}\sqrt{5}\)

Simplify: \(5\sqrt{18}-2\sqrt{8}\).

Fi àwọn àgbèwọlé hàn

	\(5\sqrt{18}-2\sqrt{8}\)
Simplify the radicals.	\(5\cdot \sqrt{9}\cdot \sqrt{2}-2\cdot \sqrt{4}\cdot \sqrt{2}\)
	\(5\cdot 3\cdot \sqrt{2}-2\cdot 2\cdot \sqrt{2}\)
	\(15\sqrt{2}-4\sqrt{2}\)
Combine the like radicals.	\(11\sqrt{2}\)

Simplify: \(4\sqrt{27}-3\sqrt{12}\).

Fi àwọn àgbèwọlé hàn

\(6\sqrt{3}\)
Simplify: \(3\sqrt{20}-7\sqrt{45}\).

Fi àwọn àgbèwọlé hàn

\(-15\sqrt{5}\)

Simplify: \(\frac{3}{4}\sqrt{192}-\frac{5}{6}\sqrt{108}\).

Fi àwọn àgbèwọlé hàn

	\(\frac{3}{4}\sqrt{192}-\frac{5}{6}\sqrt{108}\)
Simplify the radicals.	\(\frac{3}{4}\sqrt{64}\cdot \sqrt{3}-\frac{5}{6}\sqrt{36}\cdot \sqrt{3}\)
	\(\frac{3}{4}\cdot 8\cdot \sqrt{3}-\frac{5}{6}\cdot 6\cdot \sqrt{3}\)
	\(6\sqrt{3}-5\sqrt{3}\)
Combine the like radicals.	\(\sqrt{3}\)

Simplify: \(\frac{2}{3}\sqrt{108}-\frac{5}{7}\sqrt{147}\).

Fi àwọn àgbèwọlé hàn

\(\text{-}\sqrt{3}\)
Simplify: \(\frac{3}{5}\sqrt{200}-\frac{3}{4}\sqrt{128}\).

Fi àwọn àgbèwọlé hàn

\(0\)

Simplify: \(\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}\).

Fi àwọn àgbèwọlé hàn

	\(\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}\)
Simplify the radicals.	\(\frac{2}{3}\sqrt{16}\cdot \sqrt{3}-\frac{3}{4}\sqrt{4}\cdot \sqrt{3}\)
	\(\frac{2}{3}\cdot 4\cdot \sqrt{3}-\frac{3}{4}\cdot 2\cdot \sqrt{3}\)
	\(\frac{8}{3}\sqrt{3}-\frac{3}{2}\sqrt{3}\)
Find a common denominator to subtract the coefficients of the like radicals.	\(\frac{16}{6}\sqrt{3}-\frac{9}{6}\sqrt{3}\)
Simplify.	\(\frac{7}{6}\sqrt{3}\)

Simplify: \(\frac{2}{5}\sqrt{32}-\frac{1}{3}\sqrt{8}\).

Fi àwọn àgbèwọlé hàn

\(\frac{14}{15}\sqrt{2}\)
Simplify: \(\frac{1}{3}\sqrt{80}-\frac{1}{4}\sqrt{125}\).

Fi àwọn àgbèwọlé hàn

\(\frac{1}{12}\sqrt{5}\)

Simplify: \(\sqrt{18{n}^{5}}-\sqrt{32{n}^{5}}\).

Fi àwọn àgbèwọlé hàn

	\(\sqrt{18{n}^{5}}-\sqrt{32{n}^{5}}\)
Simplify the radicals.	\(\sqrt{9{n}^{4}}\cdot \sqrt{2n}-\sqrt{16{n}^{4}}\cdot \sqrt{2n}\)
	\(3{n}^{2}\sqrt{2n}-4{n}^{2}\sqrt{2n}\)
Combine the like radicals.	\(\text{-}{n}^{2}\sqrt{2n}\)

Simplify: \(\sqrt{32{m}^{7}}-\sqrt{50{m}^{7}}\).

Fi àwọn àgbèwọlé hàn

\(\text{-}{m}^{3}\sqrt{2m}\)

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Bii o ṣe le: Add and Subtract Square Roots

Add and subtract like square roots
Add and subtract square roots that need simplification
To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Àwọn Àtòjọ-ẹ̀yàn

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.

Àwọn ààyè ojú-ìwé yìí tí a fi pamọ́ láti OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Tí a fi pamọ́ síì níbẹ̀; àwọn àwọn àṣiṣe ní wa. Nítorí pé a tí fi àkóónú rẹ̀ pamọ́ fún ìlòjónú tí kò jẹ́ iṣẹ́ ọ̀fẹ́, ìwé-ìwé yìí jẹ́ ọ̀fẹ̀, kò ní ìfilọ́wọ́, àti àwọn tí a ti pamọ́ fún ìlòjónú kan.

Diẹ̀ nínú Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value