Divide Square Roots

Divide Square Roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.

Example

Try it.

Simplify: \(\frac{\sqrt{54}}{6}\).

Solution

	\(\frac{\sqrt{54}}{6}\)
Simplify the radical.	\(\frac{\sqrt{9}\cdot \sqrt{6}}{6}\)
Simplify.	\(\frac{3\sqrt{6}}{6}\)
Remove the common factors.	\(\frac{3\sqrt{6}}{3\cdot 2}\)
Simplify.	\(\frac{\sqrt{6}}{2}\)

Example

Try it.

Simplify: \(\frac{6-\sqrt{24}}{12}\).

Solution

	\(\frac{6-\sqrt{24}}{12}\)
Simplify the radical.	\(\frac{6-\sqrt{4}\cdot \sqrt{6}}{12}\)
Simplify.	\(\frac{6-2\sqrt{6}}{12}\)
Factor the common factor from the numerator.	\(\frac{2(3-\sqrt{6})}{2\cdot 6}\)
Remove the common factors.	\(\frac{2(3-\sqrt{6})}{2\cdot 6}\)
Simplify.	\(\frac{3-\sqrt{6}}{6}\)

We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says

\[\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}},b\ne 0\]

Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.

\[\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}},b\ne 0\]

We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.

We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.

Example

Try it.

Simplify: \(\frac{\sqrt{27}}{\sqrt{75}}\).

Solution

	\(\frac{\sqrt{27}}{\sqrt{75}}\)
Neither radicand is a perfect square, so rewrite using the quotient property of square roots.	\(\sqrt{\frac{27}{75}}\)
Remove common factors in the numerator and denominator.	\(\sqrt{\frac{3\cdot 9}{3\cdot 25}}\)
Simplify.	\(\sqrt{\frac{9}{25}}\)
	\(\frac{3}{5}\)

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

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.

If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.

Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.

Let’s look at a numerical example.

\(\begin{array}{llll}\text{Suppose we need an approximate value for the fraction.} & & & \frac{1}{\sqrt{2}} \\ \text{A five decimal place approximation to}\ \sqrt{2}\ \text{is}\ 1.41421. & & & \frac{1}{1.41421} \\ \text{Without a calculator, would you want to do this division?} & & & 1.414211.0\end{array}\)

But we can find a fraction equivalent to \(\frac{1}{\sqrt{2}}\) by multiplying the numerator and denominator by \(\sqrt{2}\).

Example

Try it.

Simplify: \(\frac{4}{\sqrt{3}}\).

Solution

To rationalize a denominator, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

	\(\frac{4}{\sqrt{3}}\)
Multiply both the numerator and denominator by \(\sqrt{3}.\)	\(\frac{4\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}}\)
Simplify.	\(\frac{4\sqrt{3}}{3}\)

Example

Try it.

Simplify: \(-\frac{8}{3\sqrt{6}}\).

Solution

To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by \(\sqrt{6}\).

Multiply both the numerator and the denominator by \(\sqrt{6}\).
Simplify.
Remove common factors.
Simplify.

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

Rationalize a Two-Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

\[\begin{array}{llll}(a-b)(a+b) & & & \ (2-\sqrt{5})(2+\sqrt{5}) \\ {a}^{2}-{b}^{2} & & & \ {2}^{2}-{(\sqrt{5})}^{2} \\ & & & \ 4-5 \\ & & & \ -1\end{array}\]

When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

Example

Try it.

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

Solution

Multiply the numerator and denominator by the conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.
Simplify the denominator.
Remove common factors from the numerator and denominator.
We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.

Example

Try it.

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

Solution

Multiply the numerator and denominator by the conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.
Simplify the denominator.
Simplify.

Example

Try it.

Simplify: \(\frac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}\).

Solution

Multiply the numerator and denominator by the conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.

Example

Try it.

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

Solution

Multiply the numerator and denominator by the conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.
We do not square the numerator. In factored form, we can see there are no common factors to remove from the numerator and denominator.

Key Concepts

• Quotient Property of Square Roots
  • If a, b are non-negative real numbers and \(b\ne 0\), then \[\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\ \text{and}\ \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\]
• Simplified Square Roots
  A square root is considered simplified if there are
  • no perfect square factors in the radicand
  • no fractions in the radicand
  • no square roots in the denominator of a fraction

Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

Try it.

\(\frac{\sqrt{27}}{6}\)

Solution

\(\frac{\sqrt{3}}{2}\)

Try it.

\(\frac{\sqrt{50}}{10}\)

Try it.

\(\frac{\sqrt{72}}{9}\)

Solution

\(\frac{2\sqrt{2}}{3}\)

Try it.

\(\frac{\sqrt{243}}{6}\)

Try it.

\(\frac{2-\sqrt{32}}{8}\)

Solution

\(\frac{1-2\sqrt{2}}{4}\)

Try it.

\(\frac{3+\sqrt{27}}{9}\)

Try it.

\(\frac{6+\sqrt{45}}{6}\)

Solution

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

Try it.

\(\frac{10-\sqrt{200}}{20}\)

Try it.

\(\frac{\sqrt{80}}{\sqrt{125}}\)

Solution

\(\frac{4}{5}\)

Try it.

\(\frac{\sqrt{72}}{\sqrt{200}}\)

Try it.

\(\frac{\sqrt{128}}{\sqrt{72}}\)

Solution

\(\frac{4}{3}\)

Try it.

\(\frac{\sqrt{48}}{\sqrt{75}}\)

Try it.

ⓐ \(\frac{\sqrt{8{x}^{6}}}{\sqrt{2{x}^{2}}}\) ⓑ \(\frac{\sqrt{200{m}^{5}}}{\sqrt{98m}}\)

Solution

ⓐ \(2{x}^{2}\) ⓑ \(\frac{10{m}^{2}}{7}\)

Try it.

ⓐ \(\frac{\sqrt{10{y}^{3}}}{\sqrt{5y}}\) ⓑ \(\frac{\sqrt{108{n}^{7}}}{\sqrt{243{n}^{3}}}\)

Try it.

\(\frac{\sqrt{75{r}^{3}}}{\sqrt{108r}}\)

Solution

\(\frac{5r}{6}\)

Try it.

\(\frac{\sqrt{196{q}^{5}}}{\sqrt{484q}}\)

Try it.

\(\frac{\sqrt{108{p}^{5}{q}^{2}}}{\sqrt{3{p}^{3}{q}^{6}}}\)

Solution

\(\frac{6p\sqrt{102}}{{q}^{2}}\)

Try it.

\(\frac{\sqrt{98r{s}^{10}}}{\sqrt{2{r}^{3}{s}^{4}}}\)

Try it.

\(\frac{\sqrt{320m{n}^{5}}}{\sqrt{45{m}^{7}{n}^{3}}}\)

Solution

\(\frac{8n}{3{m}^{3}}\)

Try it.

\(\frac{\sqrt{810{c}^{3}{d}^{7}}}{\sqrt{1000{c}^{5}d}}\)

Try it.

\(\frac{\sqrt{98}}{14}\)

Solution

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

Try it.

\(\frac{\sqrt{72}}{18}\)

Try it.

\(\frac{5+\sqrt{125}}{15}\)

Solution

\(\frac{1+\sqrt{5}}{3}\)

Try it.

\(\frac{6-\sqrt{45}}{12}\)

Try it.

\(\frac{\sqrt{96}}{\sqrt{150}}\)

Solution

\(\frac{4}{5}\)

Try it.

\(\frac{\sqrt{28}}{\sqrt{63}}\)

Try it.

\(\frac{\sqrt{26{y}^{7}}}{\sqrt{2y}}\)

Solution

\({y}^{3}\sqrt{13}\)

Try it.

\(\frac{\sqrt{15{x}^{3}}}{\sqrt{3x}}\)

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

Try it.

\(\frac{10}{\sqrt{6}}\)

Solution

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

Try it.

\(\frac{8}{\sqrt{3}}\)

Try it.

\(\frac{6}{\sqrt{7}}\)

Solution

\(\frac{6\sqrt{7}}{7}\)

Try it.

\(\frac{4}{\sqrt{5}}\)

Try it.

\(\frac{3}{\sqrt{13}}\)

Solution

\(\frac{3\sqrt{13}}{13}\)

Try it.

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

Try it.

\(\frac{10}{3\sqrt{10}}\)

Solution

\(\frac{\sqrt{10}}{3}\)

Try it.

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

Try it.

\(\frac{4}{9\sqrt{5}}\)

Solution

\(\frac{4\sqrt{5}}{45}\)

Try it.

\(\frac{9}{2\sqrt{7}}\)

Try it.

\(-\frac{9}{2\sqrt{3}}\)

Solution

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

Try it.

\(-\frac{8}{3\sqrt{6}}\)

Try it.

\(\sqrt{\frac{3}{20}}\)

Solution

\(\frac{\sqrt{15}}{10}\)

Try it.

\(\sqrt{\frac{4}{27}}\)

Try it.

\(\sqrt{\frac{7}{40}}\)

Solution

\(\frac{\sqrt{70}}{20}\)

Try it.

\(\sqrt{\frac{8}{45}}\)

Try it.

\(\sqrt{\frac{19}{175}}\)

Solution

\(\frac{\sqrt{133}}{35}\)

Try it.

\(\sqrt{\frac{17}{192}}\)

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

Try it.

ⓐ \(\frac{3}{3+\sqrt{11}}\) ⓑ \(\frac{8}{1-\sqrt{5}}\)

Solution

ⓐ \(\frac{3(3-\sqrt{11})}{-2}\) ⓑ \(-2(1+\sqrt{5})\)

Try it.

ⓐ \(\frac{4}{4+\sqrt{7}}\) ⓑ \(\frac{7}{2-\sqrt{6}}\)

Try it.

ⓐ \(\frac{5}{5+\sqrt{6}}\) ⓑ \(\frac{6}{3-\sqrt{7}}\)

Solution

ⓐ \(\frac{5(5-\sqrt{6})}{19}\) ⓑ \(3(3+\sqrt{7})\)

Try it.

ⓐ \(\frac{6}{6+\sqrt{5}}\) ⓑ \(\frac{5}{4-\sqrt{11}}\)

Try it.

\(\frac{\sqrt{3}}{\sqrt{m}-\sqrt{5}}\)

Solution

\(\frac{\sqrt{3}(\sqrt{m}+\sqrt{5})}{m-5}\)

Try it.

\(\frac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}\)

Try it.

\(\frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}\)

Solution

\(\frac{\sqrt{2}(\sqrt{x}+\sqrt{6})}{x-6}\)

Try it.

\(\frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}\)

Try it.

\(\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}\)

Solution

\({\frac{(\sqrt{r}+\sqrt{5})}{r-5}}^{2}\)

Try it.

\(\frac{\sqrt{s}-\sqrt{6}}{\sqrt{s}+\sqrt{6}}\)

Try it.

\(\frac{\sqrt{150{x}^{2}{y}^{6}}}{\sqrt{6{x}^{4}{y}^{2}}}\)

Solution

\(\frac{5{y}^{2}}{x}\)

Try it.

\(\frac{\sqrt{80{p}^{3}q}}{\sqrt{5p{q}^{5}}}\)

Try it.

\(\frac{15}{\sqrt{5}}\)

Solution

\(3\sqrt{5}\)

Try it.

\(\frac{3}{5\sqrt{8}}\)

Try it.

\(\sqrt{\frac{8}{54}}\)

Solution

\(\frac{2\sqrt{3}}{9}\)

Try it.

\(\sqrt{\frac{12}{20}}\)

Try it.

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

Solution

\(\frac{3(5-\sqrt{5})}{20}\)

Try it.

\(\frac{20}{4-\sqrt{3}}\)

Try it.

\(\frac{\sqrt{2}}{\sqrt{x}-\sqrt{3}}\)

Solution

\(\frac{\sqrt{2}(\sqrt{x}+\sqrt{3})}{x-3}\)

Try it.

\(\frac{\sqrt{5}}{\sqrt{y}-\sqrt{7}}\)

Try it.

\(\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}\)

Solution

\({\frac{(\sqrt{x}+2\sqrt{2})}{x-8}}^{2}\)

Try it.

\(\frac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)

Condensed — the full section is in OpenStax Elementary 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.

1. Find a fraction equivalent to \(\frac{5}{8}\) with denominator 48.
   If you missed this problem, review .

   Առաջարկել պատասխանը

   \(\frac{30}{48}\)

2. Simplify: \({(\sqrt{5})}^{2}\).
   If you missed this problem, review .

   Առաջարկել պատասխանը

   \(5\)

3. Multiply: \((7+3x)(7-3x)\).
   If you missed this problem, review .

   Առաջարկել պատասխանը

   \(49-9{x}^{2}\)

4. Simplify: \(\frac{\sqrt{54}}{6}\).

   Առաջարկել պատասխանը

   	\(\frac{\sqrt{54}}{6}\)
   Simplify the radical.	\(\frac{\sqrt{9}\cdot \sqrt{6}}{6}\)
   Simplify.	\(\frac{3\sqrt{6}}{6}\)
   Remove the common factors.	\(\frac{3\sqrt{6}}{3\cdot 2}\)
   Simplify.	\(\frac{\sqrt{6}}{2}\)

5. Simplify: \(\frac{\sqrt{32}}{8}\).

   Առաջարկել պատասխանը

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

6. Simplify: \(\frac{\sqrt{75}}{15}\).

   Առաջարկել պատասխանը

   \(\frac{\sqrt{3}}{3}\)

7. Simplify: \(\frac{6-\sqrt{24}}{12}\).

   Առաջարկել պատասխանը

   	\(\frac{6-\sqrt{24}}{12}\)
   Simplify the radical.	\(\frac{6-\sqrt{4}\cdot \sqrt{6}}{12}\)
   Simplify.	\(\frac{6-2\sqrt{6}}{12}\)
   Factor the common factor from the numerator.	\(\frac{2(3-\sqrt{6})}{2\cdot 6}\)
   Remove the common factors.	\(\frac{2(3-\sqrt{6})}{2\cdot 6}\)
   Simplify.	\(\frac{3-\sqrt{6}}{6}\)

8. Simplify: \(\frac{8-\sqrt{40}}{10}\).

   Առաջարկել պատասխանը

   \(\frac{4-\sqrt{10}}{5}\)

9. Simplify: \(\frac{10-\sqrt{75}}{20}\).

   Առաջարկել պատասխանը

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

10. Simplify: \(\frac{\sqrt{27}}{\sqrt{75}}\).

    Առաջարկել պատասխանը

    	\(\frac{\sqrt{27}}{\sqrt{75}}\)
    Neither radicand is a perfect square, so rewrite using the quotient property of square roots.	\(\sqrt{\frac{27}{75}}\)
    Remove common factors in the numerator and denominator.	\(\sqrt{\frac{3\cdot 9}{3\cdot 25}}\)
    Simplify.	\(\sqrt{\frac{9}{25}}\)
    	\(\frac{3}{5}\)

11. Simplify: \(\frac{\sqrt{48}}{\sqrt{108}}\).

    Առաջարկել պատասխանը

    \(\frac{2}{3}\)

12. Simplify: \(\frac{\sqrt{96}}{\sqrt{54}}\).

    Առաջարկել պատասխանը

    \(\frac{4}{3}\)

13. Simplify: \(\frac{\sqrt{6{y}^{5}}}{\sqrt{2y}}\).

    Առաջարկել պատասխանը

    	\(\frac{\sqrt{6{y}^{5}}}{\sqrt{2y}}\)
    Neither radicand is a perfect square, so rewrite using the quotient property of square roots.	\(\sqrt{\frac{6{y}^{5}}{2y}}\)
    Remove common factors in the numerator and denominator.	\(\sqrt{\frac{2\cdot 3\cdot {y}^{4}\cdot y}{2\cdot y}}\)
    Simplify.	\(\sqrt{3{y}^{4}}\)
    Simplify the radical.	\({y}^{2}\sqrt{3}\)

14. Simplify: \(\frac{\sqrt{12{r}^{3}}}{\sqrt{6r}}\).

    Առաջարկել պատասխանը

    \(r\sqrt{2}\)

15. Simplify: \(\frac{\sqrt{14{p}^{9}}}{\sqrt{2{p}^{5}}}\).

    Առաջարկել պատասխանը

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

16. Simplify: \(\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}\).

    Առաջարկել պատասխանը

    	\(\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}\)
    Rewrite using the quotient property of square roots.	\(\sqrt{\frac{72{x}^{3}}{162x}}\)
    Remove common factors.	\(\sqrt{\frac{18\cdot 4\cdot {x}^{2}\cdot x}{18\cdot 9\cdot x}}\)
    Simplify.	\(\sqrt{\frac{4{x}^{2}}{9}}\)
    Simplify the radical.	\(\frac{2x}{3}\)

17. Simplify: \(\frac{\sqrt{50{s}^{3}}}{\sqrt{128s}}\).

    Առաջարկել պատասխանը

    \(\frac{5s}{8}\)

18. Simplify: \(\frac{\sqrt{75{q}^{5}}}{\sqrt{108q}}\).

    Առաջարկել պատասխանը

    \(\frac{5{q}^{2}}{6}\)

19. Simplify: \(\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}\).

    Առաջարկել պատասխանը

    	\(\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}\)
    Rewrite using the quotient property of square roots.	\(\sqrt{\frac{147a{b}^{8}}{3{a}^{3}{b}^{4}}}\)
    Remove common factors.	\(\sqrt{\frac{49{b}^{4}}{{a}^{2}}}\)
    Simplify the radical.	\(\frac{7{b}^{2}}{a}\)

20. Simplify: \(\frac{\sqrt{162{x}^{10}{y}^{2}}}{\sqrt{2{x}^{6}{y}^{6}}}\).

    Առաջարկել պատասխանը

    \(\frac{9{x}^{2}}{{y}^{2}}\)

21. Simplify: \(\frac{\sqrt{300{m}^{3}{n}^{7}}}{\sqrt{3{m}^{5}n}}\).

    Առաջարկել պատասխանը

    \(\frac{10{n}^{3}}{m}\)

22. Simplify: \(\frac{4}{\sqrt{3}}\).

    Առաջարկել պատասխանը

    To rationalize a denominator, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

    	\(\frac{4}{\sqrt{3}}\)
    Multiply both the numerator and denominator by \(\sqrt{3}.\)	\(\frac{4\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}}\)
    Simplify.	\(\frac{4\sqrt{3}}{3}\)

23. Simplify: \(\frac{5}{\sqrt{3}}\).

    Առաջարկել պատասխանը

    \(\frac{5\sqrt{3}}{3}\)

24. Simplify: \(\frac{6}{\sqrt{5}}\).

    Առաջարկել պատասխանը

    \(\frac{6\sqrt{5}}{5}\)

25. Simplify: \(-\frac{8}{3\sqrt{6}}\).

    Առաջարկել պատասխանը

    To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by \(\sqrt{6}\).

    Multiply both the numerator and the denominator by \(\sqrt{6}\).
    Simplify.
    Remove common factors.
    Simplify.

26. Simplify: \(\frac{5}{2\sqrt{5}}\).

    Առաջարկել պատասխանը

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

27. Simplify: \(-\frac{9}{4\sqrt{3}}\).

    Առաջարկել պատասխանը

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

28. Simplify: \(\sqrt{\frac{5}{12}}\).

    Առաջարկել պատասխանը

    The fraction is not a perfect square, so rewrite using the Quotient Property.
    Simplify the denominator
    Rationalize the denominator.
    Simplify.
    Simplify.

29. Simplify: \(\sqrt{\frac{7}{18}}\).

    Առաջարկել պատասխանը

    \(\frac{\sqrt{14}}{6}\)

30. Simplify: \(\sqrt{\frac{3}{32}}\).

    Առաջարկել պատասխանը

    \(\frac{\sqrt{6}}{8}\)

31. Simplify: \(\sqrt{\frac{11}{28}}\).

    Առաջարկել պատասխանը

    Rewrite using the Quotient Property.
    Simplify the denominator.
    Rationalize the denominator.
    Simplify.
    Simplify.

32. Simplify: \(\sqrt{\frac{3}{27}}\).

    Առաջարկել պատասխանը

    \(\frac{1}{3}\)

33. Simplify: \(\sqrt{\frac{10}{50}}\).

    Առաջարկել պատասխանը

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

34. Simplify: \(\frac{4}{4+\sqrt{2}}\).

    Առաջարկել պատասխանը

    Multiply the numerator and denominator by the conjugate of the denominator.
    Multiply the conjugates in the denominator.
    Simplify the denominator.
    Simplify the denominator.
    Remove common factors from the numerator and denominator.
    We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.

35. Simplify: \(\frac{2}{2+\sqrt{3}}\).

    Առաջարկել պատասխանը

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

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

    Առաջարկել պատասխանը

    \(\frac{5(5-\sqrt{3})}{22}\)

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

    Առաջարկել պատասխանը

    Multiply the numerator and denominator by the conjugate of the denominator.
    Multiply the conjugates in the denominator.
    Simplify the denominator.
    Simplify the denominator.
    Simplify.

38. Simplify: \(\frac{3}{1-\sqrt{5}}\).

    Առաջարկել պատասխանը

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

39. Simplify: \(\frac{2}{4-\sqrt{6}}\).

    Առաջարկել պատասխանը

    \(\frac{4+\sqrt{6}}{5}\)

40. Simplify: \(\frac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}\).

    Առաջարկել պատասխանը

    Multiply the numerator and denominator by the conjugate of the denominator.
    Multiply the conjugates in the denominator.
    Simplify the denominator.

Symbols used here

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

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

\neq

not equal
The two sides are different.

\pm

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

\leq,\ \geq

less/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.

i

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: Divide Square Roots

1. Divide square roots
2. Rationalize a one-term denominator
3. Rationalize a two-term denominator
4. no perfect-square factors in the radicand
5. no fractions in the radicand
6. no square roots in the denominator of a fraction
7. If
8. no perfect square factors in the radicand

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.

Փորձեք ինքներդ

⌨ Type ✎ Draw

Ձեր մաթեմատիկական խնդիրը

Լրացնել

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

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

Ցուցադրել Algebra

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