Add and Subtract 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.

Ankehitriny ianao Tsy misy milina fanisana afaka mamaha io olana io, saingy azo atao ny maminavina ny ampahany amin'io. Andramo ny iray eto ambany, na soraty ny anao manokana.

⌨ Karazana ✎ Sary

Ny olanao amin'ny matematika

π _Soloy

Fanazaran-tena (40)

Andramo amin'ny taratasy aloha ny tsirairay. Asehoy ny valinteny mba hijerena; azo sokafana ao amin'ny mpamolavola ny valinteny voaporofo ho an'ny dingana tsirairay.

1. Add: ⓐ \(3x+9x\) ⓑ \(5m+5n\).

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

   ⓐ \(12x\) ⓑ \(5m+5n\)

2. Simplify: \(\sqrt{50{x}^{3}}\).

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

   \(5x\sqrt{2x}\)

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

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

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

4. Simplify: \(8\sqrt{2}-9\sqrt{2}\).

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

   \(\text{-}\sqrt{2}\)

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

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

   \(-4\sqrt{3}\)

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

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

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

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

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

   \(9\sqrt{x}\)

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

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

   \(8\sqrt{u}\)

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

   Miasa amin'izany Toro-hevitra

   Asehoy ny valinteny

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

10. Simplify: \(7\sqrt{p}-6\sqrt{q}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(7\sqrt{p}-6\sqrt{q}\)

11. Simplify: \(6\sqrt{a}-3\sqrt{b}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(6\sqrt{a}-3\sqrt{b}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

13. Simplify: \(4\sqrt{11}+2\sqrt{11}+3\sqrt{11}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(9\sqrt{11}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(11\sqrt{10}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(-5\sqrt{7}+2\sqrt{5}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

19. Simplify: \(\sqrt{7x}-7\sqrt{7x}+4\sqrt{7x}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

23. Simplify: \(3\sqrt{7mn}+\sqrt{7mn}-4\sqrt{7mn}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(0\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

25. Simplify: \(\sqrt{18}+6\sqrt{2}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(9\sqrt{2}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(7\sqrt{3}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

28. Simplify: \(\sqrt{32}-\sqrt{18}\).

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(\sqrt{2}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    	\(5\sqrt{18}-2\sqrt{8}\)
    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.	\(11\sqrt{2}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(6\sqrt{3}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(-15\sqrt{5}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(0\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    	\(\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}\)
    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.	\(\frac{7}{6}\sqrt{3}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    \(\frac{14}{15}\sqrt{2}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

    	\(\sqrt{18{n}^{5}}-\sqrt{32{n}^{5}}\)
    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.	\(\text{-}{n}^{2}\sqrt{2n}\)

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

    Miasa amin'izany Toro-hevitra

    Asehoy ny valinteny

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

Aoka ho miasa ny anao

Ny kaonty maimaimpoana dia mampiditra ny fanamarihana amin'ny lesona tsirairay, ny tahirin'ny zavatra vitanao, ny olana voavahao ao amin'ny toerana iray, ary ny mpampianatra iray izay azonao anontanio momba ity pejy ity. misokatra ho an'ny rehetra ny matematika, na efa niditra na tsy niditra.

Misoratra anarana Fidirana

Marika ampiasaina eto

Tsindrio ny marika iray mba hahitana ny famaritana feno, sary, ary ny dikan'ny litera tsirairay ao aminy.

\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Ahoana ny fomba: Add and Subtract Square Roots

1. Add and subtract like square roots
2. Add and subtract square roots that need simplification
3. To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
4. 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.

Fanontaniana napetrak'ireo olona

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.

Nalaina avy amin'ny OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Nofafana sy nanazavana indray eto; ny hadisoana dia antsika. Satria nahazo alalana ho an'ny fampiasana tsy ara-barotra ny loharanom-baovaony, maimaimpoana ity lesona ity, tsy mivarotra na inona na inona, ary mizara amin'ny alalana mitovy.

Mbola maro ao Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value