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Divide Radical Expressions

Divide radical expressions

Divide Radical Expressions

We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals.

We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars are needed.

We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.

Example

Try it.

Simplify: ⓐ \(\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}\) ⓑ \(\frac{\sqrt[3]{32{x}^{2}}}{\sqrt[3]{4{x}^{5}}}.\)

Solution

ⓐ

	\(\ \frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}\)
Rewrite using the quotient property, \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}.\)	\(\ \sqrt{\frac{72{x}^{3}}{162x}}\)
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.	\(\ \frac{2x}{3}\)

ⓑ

	\(\ \frac{\sqrt[3]{32{x}^{2}}}{\sqrt[3]{4{x}^{5}}}\)
Rewrite using the quotient property, \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}.\)	\(\ \sqrt[3]{\frac{32{x}^{2}}{4{x}^{5}}}\)
Simplify the fraction under the radical.	\(\ \sqrt[3]{\frac{8}{{x}^{3}}}\)
Simplify the radical.	\(\ \frac{2}{x}\)

Example

Try it.

Simplify: ⓐ \(\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}\) ⓑ \(\frac{\sqrt[3]{-250{m}^{}{n}^{-2}}}{\sqrt[3]{2{m}^{-2}{n}^{4}}}.\)

Solution

ⓐ

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

ⓑ

	\(\ \frac{\sqrt[3]{-250{m}^{}{n}^{-2}}}{\sqrt[3]{2{m}^{-2}{n}^{4}}}\)
Rewrite using the quotient property.	\(\ \sqrt[3]{\frac{-250{m}^{}{n}^{-2}}{2{m}^{-2}{n}^{4}}}\)
Simplify the fraction under the radical.	\(\ \sqrt[3]{\frac{-125{m}^{3}}{{n}^{6}}}\)
Simplify the radical.	\(\ -\frac{5m}{{n}^{2}}\)

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

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator 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. 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.

This process is still used today, and is useful in other areas of mathematics, too.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a radical.

Similarly, a radical expression is not considered simplified if the radicand contains a fraction.

To rationalize a denominator with a square root, we use the property that \({(\sqrt{a})}^{2}=a.\) If we square an irrational square root, we get a rational number.

We will use this property to rationalize the denominator in the next example.

Example

Try it.

Simplify: ⓐ \(\frac{4}{\sqrt{3}}\) ⓑ \(\sqrt{\frac{3}{20}}\) ⓒ \(\frac{3}{\sqrt{6x}}.\)

Solution

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

ⓐ


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

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect square, so rewrite using the Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by \(\sqrt{5}.\)
Simplify.
Simplify.

ⓒ


Multiply the numerator and denominator by \(\sqrt{6x}.\)
Simplify.
Simplify.

Condensed — the full section is in OpenStax Intermediate 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: \(\frac{5}{2-\sqrt{3}}.\)

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.

Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.

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.

Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.

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. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.

Key Concepts

Quotient Property of Radical Expressions
- If \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\) are real numbers, \(b\ne 0,\) and for any integer \(n\ge 2\) then,
  \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) and \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\)
Simplified Radical Expressions
- A radical expression is considered simplified if there are:
  - no factors in the radicand that have perfect powers of the index
  - no fractions in the radicand
  - no radicals in the denominator of a fraction

Divide Radical Expressions

Divide Square Roots

In the following exercises, simplify.

Try it.

ⓐ \(\frac{\sqrt{128}}{\sqrt{72}}\) ⓑ \(\frac{\sqrt[3]{128}}{\sqrt[3]{54}}\)

Solution

ⓐ \(\frac{4}{3}\) ⓑ \(\frac{4}{3}\)

Try it.

ⓐ \(\frac{\sqrt{48}}{\sqrt{75}}\) ⓑ \(\frac{\sqrt[3]{81}}{\sqrt[3]{24}}\)

Try it.

ⓐ \(\frac{\sqrt{200{m}^{5}}}{\sqrt{98m}}\) ⓑ \(\frac{\sqrt[3]{54{y}^{2}}}{\sqrt[3]{2{y}^{5}}}\)

Solution

ⓐ \(\frac{10{m}^{2}}{7}\) ⓑ \(\frac{3}{y}\)

Try it.

ⓐ \(\frac{\sqrt{108{n}^{7}}}{\sqrt{243{n}^{3}}}\) ⓑ \(\frac{\sqrt[3]{54{y}^{}}}{\sqrt[3]{16{y}^{4}}}\)

Try it.

ⓐ \(\frac{\sqrt{75{r}^{3}}}{\sqrt{108{r}^{7}}}\) ⓑ \(\frac{\sqrt[3]{24{x}^{7}}}{\sqrt[3]{81{x}^{4}}}\)

Solution

ⓐ \(\frac{5}{6{r}^{2}}\) ⓑ \(\frac{2x}{3}\)

Try it.

ⓐ \(\frac{\sqrt{196{q}^{}}}{\sqrt{484{q}^{5}}}\) ⓑ \(\frac{\sqrt[3]{16{m}^{4}}}{\sqrt[3]{54{m}^{}}}\)

Try it.

ⓐ \(\frac{\sqrt{108{p}^{5}{q}^{2}}}{\sqrt{3{p}^{3}{q}^{6}}}\) ⓑ \(\frac{\sqrt[3]{-16{a}^{4}{b}^{-2}}}{\sqrt[3]{2{a}^{-2}{b}^{}}}\)

Solution

ⓐ \(\frac{6p}{{q}^{2}}\) ⓑ \(-\frac{2{a}^{2}}{b}\)

Try it.

ⓐ \(\frac{\sqrt{98r{s}^{10}}}{\sqrt{2{r}^{3}{s}^{4}}}\) ⓑ \(\frac{\sqrt[3]{-375{y}^{4}{z}^{-2}}}{\sqrt[3]{3{y}^{-2}{z}^{4}}}\)

Try it.

ⓐ \(\frac{\sqrt{320m{n}^{-5}}}{\sqrt{45{m}^{-7}{n}^{3}}}\) ⓑ \(\frac{\sqrt[3]{16{x}^{4}{y}^{-2}}}{\sqrt[3]{-54{x}^{-2}{y}^{4}}}\)

Solution

ⓐ \(\frac{8{m}^{4}}{3{n}^{4}}\) ⓑ \(-\frac{2{x}^{2}}{3{y}^{2}}\)

Try it.

ⓐ \(\frac{\sqrt{810{c}^{-3}{d}^{7}}}{\sqrt{1000{c}^{}{d}^{-1}}}\) ⓑ \(\frac{\sqrt[3]{24{a}^{7}{b}^{-1}}}{\sqrt[3]{-81{a}^{-2}{b}^{2}}}\)

Try it.

\(\frac{\sqrt{56{x}^{5}{y}^{4}}}{\sqrt{2x{y}^{3}}}\)

Solution

\(2{x}^{2}\sqrt{7y}\)

Try it.

\(\frac{\sqrt{72{a}^{3}{b}^{6}}}{\sqrt{3a{b}^{3}}}\)

Try it.

\(\frac{\sqrt[3]{48{a}^{3}{b}^{6}}}{\sqrt[3]{3{a}^{-1}{b}^{3}}}\)

Solution

\(2ab\sqrt[3]{2a}\)

Try it.

\(\frac{\sqrt[3]{162{x}^{-3}{y}^{6}}}{\sqrt[3]{2{x}^{3}{y}^{-2}}}\)

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Try it.

ⓐ \(\frac{10}{\sqrt{6}}\) ⓑ \(\sqrt{\frac{4}{27}}\) ⓒ \(\frac{10}{\sqrt{5x}}\)

Solution

ⓐ \(\frac{5\sqrt{6}}{3}\) ⓑ \(\frac{2\sqrt{3}}{9}\) ⓒ \(\frac{2\sqrt{5x}}{x}\)

Try it.

ⓐ \(\frac{8}{\sqrt{3}}\) ⓑ \(\sqrt{\frac{7}{40}}\) ⓒ \(\frac{8}{\sqrt{2y}}\)

Try it.

ⓐ \(\frac{6}{\sqrt{7}}\) ⓑ \(\sqrt{\frac{8}{45}}\) ⓒ \(\frac{12}{\sqrt{3p}}\)

Solution

ⓐ \(\frac{6\sqrt{7}}{7}\) ⓑ \(\frac{2\sqrt{10}}{15}\) ⓒ \(\frac{4\sqrt{3p}}{p}\)

Try it.

ⓐ \(\frac{4}{\sqrt{5}}\) ⓑ \(\sqrt{\frac{27}{80}}\) ⓒ \(\frac{18}{\sqrt{6q}}\)

Try it.

ⓐ \(\frac{1}{\sqrt[3]{5}}\) ⓑ \(\sqrt[3]{\frac{5}{24}}\) ⓒ \(\frac{4}{\sqrt[3]{36a}}\)

Solution

ⓐ \(\frac{\sqrt[3]{25}}{5}\) ⓑ \(\frac{\sqrt[3]{45}}{6}\) ⓒ \(\frac{2\sqrt[3]{6{a}^{2}}}{3a}\)

Try it.

ⓐ \(\frac{1}{\sqrt[3]{3}}\) ⓑ \(\sqrt[3]{\frac{5}{32}}\) ⓒ \(\frac{7}{\sqrt[3]{49b}}\)

Try it.

ⓐ \(\frac{1}{\sqrt[3]{11}}\) ⓑ \(\sqrt[3]{\frac{7}{54}}\) ⓒ \(\frac{3}{\sqrt[3]{3{x}^{2}}}\)

Solution

ⓐ \(\frac{\sqrt[3]{121}}{11}\) ⓑ \(\frac{\sqrt[3]{28}}{6}\) ⓒ \(\frac{\sqrt[3]{9x}}{x}\)

Try it.

ⓐ \(\frac{1}{\sqrt[3]{13}}\) ⓑ \(\sqrt[3]{\frac{3}{128}}\) ⓒ \(\frac{3}{\sqrt[3]{6{y}^{2}}}\)

Try it.

ⓐ \(\frac{1}{\sqrt[4]{7}}\) ⓑ \(\sqrt[4]{\frac{5}{32}}\) ⓒ \(\frac{4}{\sqrt[4]{4{x}^{2}}}\)

Solution

ⓐ \(\frac{\sqrt[4]{343}}{7}\) ⓑ \(\frac{\sqrt[4]{40}}{4}\) ⓒ \(\frac{2\sqrt[4]{4{x}^{2}}}{x}\)

Try it.

ⓐ \(\frac{1}{\sqrt[4]{4}}\) ⓑ \(\sqrt[4]{\frac{9}{32}}\) ⓒ \(\frac{6}{\sqrt[4]{9{x}^{3}}}\)

Try it.

ⓐ \(\frac{1}{\sqrt[4]{9}}\) ⓑ \(\sqrt[4]{\frac{25}{128}}\) ⓒ \(\frac{6}{\sqrt[4]{27a}}\)

Solution

ⓐ \(\frac{\sqrt[4]{9}}{3}\) ⓑ \(\frac{\sqrt[4]{50}}{4}\) ⓒ \(\frac{2\sqrt[4]{3{a}^{3}}}{a}\)

Try it.

ⓐ \(\frac{1}{\sqrt[4]{8}}\) ⓑ \(\sqrt[4]{\frac{27}{128}}\) ⓒ \(\frac{16}{\sqrt[4]{64{b}^{2}}}\)

Rationalize a Two Term Denominator

In the following exercises, simplify.

Try it.

\(\frac{8}{1-\sqrt{5}}\)

Solution

\(-2(1+\sqrt{5})\)

Try it.

\(\frac{7}{2-\sqrt{6}}\)

Try it.

\(\frac{6}{3-\sqrt{7}}\)

Solution

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

Try it.

\(\frac{5}{4-\sqrt{11}}\)

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.

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

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 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.

Simplify: \(\frac{30}{48}.\)
If you missed this problem, review .

Одкриј го одговорот

\(\frac{5}{8}\)
Simplify: \({x}^{2}\cdot {x}^{4}.\)
If you missed this problem, review .

Одкриј го одговорот

\({x}^{6}\)
Multiply: \((7+3x)(7-3x).\)
If you missed this problem, review .

Одкриј го одговорот

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

Simplify: ⓐ \(\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}\) ⓑ \(\frac{\sqrt[3]{32{x}^{2}}}{\sqrt[3]{4{x}^{5}}}.\)

Одкриј го одговорот

ⓐ

	\(\ \frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}\)
Rewrite using the quotient property, \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}.\)	\(\ \sqrt{\frac{72{x}^{3}}{162x}}\)
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.	\(\ \frac{2x}{3}\)

ⓑ

	\(\ \frac{\sqrt[3]{32{x}^{2}}}{\sqrt[3]{4{x}^{5}}}\)
Rewrite using the quotient property, \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}.\)	\(\ \sqrt[3]{\frac{32{x}^{2}}{4{x}^{5}}}\)
Simplify the fraction under the radical.	\(\ \sqrt[3]{\frac{8}{{x}^{3}}}\)
Simplify the radical.	\(\ \frac{2}{x}\)

Simplify: ⓐ \(\frac{\sqrt{50{s}^{3}}}{\sqrt{128s}}\) ⓑ \(\frac{\sqrt[3]{56{a}^{}}}{\sqrt[3]{7{a}^{4}}}.\)

Одкриј го одговорот

ⓐ \(\frac{5s}{8}\) ⓑ \(\frac{2}{a}\)
Simplify: ⓐ \(\frac{\sqrt{75{q}^{5}}}{\sqrt{108q}}\) ⓑ \(\frac{\sqrt[3]{72{b}^{2}}}{\sqrt[3]{9{b}^{5}}}.\)

Одкриј го одговорот

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

Simplify: ⓐ \(\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}\) ⓑ \(\frac{\sqrt[3]{-250{m}^{}{n}^{-2}}}{\sqrt[3]{2{m}^{-2}{n}^{4}}}.\)

Одкриј го одговорот

ⓐ

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

ⓑ

	\(\ \frac{\sqrt[3]{-250{m}^{}{n}^{-2}}}{\sqrt[3]{2{m}^{-2}{n}^{4}}}\)
Rewrite using the quotient property.	\(\ \sqrt[3]{\frac{-250{m}^{}{n}^{-2}}{2{m}^{-2}{n}^{4}}}\)
Simplify the fraction under the radical.	\(\ \sqrt[3]{\frac{-125{m}^{3}}{{n}^{6}}}\)
Simplify the radical.	\(\ -\frac{5m}{{n}^{2}}\)

Simplify: ⓐ \(\frac{\sqrt{162{x}^{10}{y}^{2}}}{\sqrt{2{x}^{6}{y}^{6}}}\) ⓑ \(\frac{\sqrt[3]{-128{x}^{2}{y}^{-1}}}{\sqrt[3]{2{x}^{-1}{y}^{2}}}.\)

Одкриј го одговорот

ⓐ \(\frac{9{x}^{2}}{{y}^{2}}\) ⓑ \(\frac{-4x}{y}\)
Simplify: ⓐ \(\frac{\sqrt{300{m}^{3}{n}^{7}}}{\sqrt{3{m}^{5}n}}\) ⓑ \(\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3{p}^{-2}{q}^{5}}}.\)

Одкриј го одговорот

ⓐ \(\frac{10{n}^{3}}{m}\) ⓑ \(\frac{-3p}{{q}^{2}}\)

Simplify: \(\frac{\sqrt{54{x}^{5}{y}^{3}}}{\sqrt{3{x}^{2}y}}.\)

Одкриј го одговорот

	\(\ \frac{\sqrt{54{x}^{5}{y}^{3}}}{\sqrt{3{x}^{2}y}}\)
Rewrite using the quotient property.	\(\ \sqrt{\frac{54{x}^{5}{y}^{3}}{3{x}^{2}y}}\)
Remove common factors in the fraction.	\(\ \sqrt{18{x}^{3}{y}^{2}}\)
Rewrite the radicand as a product using the largest perfect square factor.	\(\ \sqrt{9{x}^{2}{y}^{2}⋅2x}\)
Rewrite the radical as the product of two radicals.	\(\ \sqrt{9{x}^{2}{y}^{2}}⋅\sqrt{2x}\)
Simplify.	\(\ 3xy\sqrt{2x}\)

Simplify: \(\frac{\sqrt{64{x}^{4}{y}^{5}}}{\sqrt{2x{y}^{3}}}.\)

Одкриј го одговорот

\(4xy\sqrt{2x}\)
Simplify: \(\frac{\sqrt{96{a}^{5}{b}^{4}}}{\sqrt{2{a}^{3}b}}.\)

Одкриј го одговорот

\(4ab\sqrt{3b}\)
Simplify: ⓐ \(\frac{4}{\sqrt{3}}\) ⓑ \(\sqrt{\frac{3}{20}}\) ⓒ \(\frac{3}{\sqrt{6x}}.\)

Одкриј го одговорот

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

Multiply both the numerator and denominator by \(\sqrt{3}.\)
Simplify.
ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.
The fraction is not a perfect square, so rewrite using the
Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by \(\sqrt{5}.\)
Simplify.
Simplify.
ⓒ

Multiply the numerator and denominator by \(\sqrt{6x}.\)
Simplify.
Simplify.
Simplify: ⓐ \(\frac{5}{\sqrt{3}}\) ⓑ \(\sqrt{\frac{3}{32}}\) ⓒ \(\frac{2}{\sqrt{2x}}.\)

Одкриј го одговорот

ⓐ \(\frac{5\sqrt{3}}{3}\) ⓑ \(\frac{\sqrt{6}}{8}\) ⓒ \(\frac{\sqrt{2x}}{x}\)
Simplify: ⓐ \(\frac{6}{\sqrt{5}}\) ⓑ \(\sqrt{\frac{7}{18}}\) ⓒ \(\frac{5}{\sqrt{5x}}.\)

Одкриј го одговорот

ⓐ \(\frac{6\sqrt{5}}{5}\) ⓑ \(\frac{\sqrt{14}}{6}\) ⓒ \(\frac{\sqrt{5x}}{x}\)

Одкриј го одговорот

To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ


The radical in the denominator has one factor of 6. Multiply both the numerator and denominator by \(\sqrt[3]{{6}^{2}},\) which gives us 2 more factors of 6.
Multiply. Notice the radicand in the denominator has 3 powers of 6.
Simplify the cube root in the denominator.

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect cube, so rewrite using the Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by \(\sqrt[3]{{3}^{2}}.\) This will give us 3 factors of 3.
Simplify.
Remember, \(\sqrt[3]{{3}^{3}}=3.\)
Simplify.

ⓒ


Rewrite the radicand to show the factors.
Multiply the numerator and denominator by \(\sqrt[3]{2\cdot {x}^{2}}.\) This will get us 3 factors of 2 and 3 factors of x.
Simplify.
Simplify the radical in the denominator.

Simplify: ⓐ \(\frac{1}{\sqrt[3]{7}}\) ⓑ \(\sqrt[3]{\frac{5}{12}}\) ⓒ \(\frac{5}{\sqrt[3]{9y}}.\)

Одкриј го одговорот

ⓐ \(\frac{\sqrt[3]{49}}{7}\) ⓑ \(\frac{\sqrt[3]{90}}{6}\) ⓒ \(\frac{5\sqrt[3]{3{y}^{2}}}{3y}\)
Simplify: ⓐ \(\frac{1}{\sqrt[3]{2}}\) ⓑ \(\sqrt[3]{\frac{3}{20}}\) ⓒ \(\frac{2}{\sqrt[3]{25n}}.\)

Одкриј го одговорот

ⓐ \(\frac{\sqrt[3]{4}}{2}\) ⓑ \(\frac{\sqrt[3]{150}}{10}\) ⓒ \(\frac{2\sqrt[3]{5{n}^{2}}}{5n}\)

Одкриј го одговорот

To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ


The radical in the denominator has one factor of 2. Multiply both the numerator and denominator by \(\sqrt[4]{{2}^{3}},\) which gives us 3 more factors of 2.
Multiply. Notice the radicand in the denominator has 4 powers of 2.
Simplify the fourth root in the denominator.

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect fourth power, so rewrite using the Quotient Property.
Rewrite the radicand in the denominator to show the factors.
Simplify the denominator.
Multiply the numerator and denominator by \(\sqrt[4]{{2}^{2}}.\) This will give us 4 factors of 2.
Simplify.
Remember, \(\sqrt[4]{{2}^{4}}=2.\)
Simplify.

ⓒ


Rewrite the radicand to show the factors.
Multiply the numerator and denominator by \(\sqrt[4]{2\cdot {x}^{3}}.\) This will get us 4 factors of 2 and 4 factors of x.
Simplify.
Simplify the radical in the denominator.
Simplify the fraction.

Simplify: ⓐ \(\frac{1}{\sqrt[4]{3}}\) ⓑ \(\sqrt[4]{\frac{3}{64}}\) ⓒ \(\frac{3}{\sqrt[4]{125x}}.\)

Одкриј го одговорот

ⓐ \(\frac{\sqrt[4]{27}}{3}\) ⓑ \(\frac{\sqrt[4]{12}}{4}\) ⓒ \(\frac{3\sqrt[4]{5{x}^{3}}}{5x}\)
Simplify: ⓐ \(\frac{1}{\sqrt[4]{5}}\) ⓑ \(\sqrt[4]{\frac{7}{128}}\) ⓒ \(\frac{4}{\sqrt[4]{4x}}\)

Одкриј го одговорот

ⓐ \(\frac{\sqrt[4]{125}}{5}\) ⓑ \(\frac{\sqrt[4]{14}}{4}\)
ⓒ \(\frac{2\sqrt[4]{4{x}^{3}}}{x}\)
Simplify: \(\frac{5}{2-\sqrt{3}}.\)

Одкриј го одговорот

Multiply the numerator and denominator by the
conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.
Simplify the denominator.
Simplify.
Simplify: \(\frac{3}{1-\sqrt{5}}.\)

Одкриј го одговорот

\(-\frac{3(1+\sqrt{5})}{4}\)
Simplify: \(\frac{2}{4-\sqrt{6}}.\)

Одкриј го одговорот

\(\frac{4+\sqrt{6}}{5}\)
Simplify: \(\frac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}.\)

Одкриј го одговорот

Multiply the numerator and denominator by the
conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.
Simplify: \(\frac{\sqrt{5}}{\sqrt{x}+\sqrt{2}}.\)

Одкриј го одговорот

\(\frac{\sqrt{5}(\sqrt{x}-\sqrt{2})}{x-2}\)
Simplify: \(\frac{\sqrt{10}}{\sqrt{y}-\sqrt{3}}.\)

Одкриј го одговорот

\(\frac{\sqrt{10}(\sqrt{y}+\sqrt{3})}{y-3}\)
Simplify: \(\frac{\sqrt{x}+\sqrt{7}}{\sqrt{x}-\sqrt{7}}.\)

Одкриј го одговорот

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. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.
Simplify: \(\frac{\sqrt{p}+\sqrt{2}}{\sqrt{p}-\sqrt{2}}.\)

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\({\frac{(\sqrt{p}+\sqrt{2})}{p-2}}^{2}\)
Simplify: \(\frac{\sqrt{q}-\sqrt{10}}{\sqrt{q}+\sqrt{10}}\)

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\({\frac{(\sqrt{q}-\sqrt{10})}{q-10}}^{2}\)
ⓐ \(\frac{\sqrt{128}}{\sqrt{72}}\) ⓑ \(\frac{\sqrt[3]{128}}{\sqrt[3]{54}}\)

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ⓐ \(\frac{4}{3}\) ⓑ \(\frac{4}{3}\)
ⓐ \(\frac{\sqrt{48}}{\sqrt{75}}\) ⓑ \(\frac{\sqrt[3]{81}}{\sqrt[3]{24}}\)
ⓐ \(\frac{\sqrt{200{m}^{5}}}{\sqrt{98m}}\) ⓑ \(\frac{\sqrt[3]{54{y}^{2}}}{\sqrt[3]{2{y}^{5}}}\)

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ⓐ \(\frac{10{m}^{2}}{7}\) ⓑ \(\frac{3}{y}\)
ⓐ \(\frac{\sqrt{108{n}^{7}}}{\sqrt{243{n}^{3}}}\) ⓑ \(\frac{\sqrt[3]{54{y}^{}}}{\sqrt[3]{16{y}^{4}}}\)
ⓐ \(\frac{\sqrt{75{r}^{3}}}{\sqrt{108{r}^{7}}}\) ⓑ \(\frac{\sqrt[3]{24{x}^{7}}}{\sqrt[3]{81{x}^{4}}}\)

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ⓐ \(\frac{5}{6{r}^{2}}\) ⓑ \(\frac{2x}{3}\)
ⓐ \(\frac{\sqrt{196{q}^{}}}{\sqrt{484{q}^{5}}}\) ⓑ \(\frac{\sqrt[3]{16{m}^{4}}}{\sqrt[3]{54{m}^{}}}\)
ⓐ \(\frac{\sqrt{108{p}^{5}{q}^{2}}}{\sqrt{3{p}^{3}{q}^{6}}}\) ⓑ \(\frac{\sqrt[3]{-16{a}^{4}{b}^{-2}}}{\sqrt[3]{2{a}^{-2}{b}^{}}}\)

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ⓐ \(\frac{6p}{{q}^{2}}\) ⓑ \(-\frac{2{a}^{2}}{b}\)
ⓐ \(\frac{\sqrt{98r{s}^{10}}}{\sqrt{2{r}^{3}{s}^{4}}}\) ⓑ \(\frac{\sqrt[3]{-375{y}^{4}{z}^{-2}}}{\sqrt[3]{3{y}^{-2}{z}^{4}}}\)
ⓐ \(\frac{\sqrt{320m{n}^{-5}}}{\sqrt{45{m}^{-7}{n}^{3}}}\) ⓑ \(\frac{\sqrt[3]{16{x}^{4}{y}^{-2}}}{\sqrt[3]{-54{x}^{-2}{y}^{4}}}\)

Одкриј го одговорот

ⓐ \(\frac{8{m}^{4}}{3{n}^{4}}\) ⓑ \(-\frac{2{x}^{2}}{3{y}^{2}}\)
ⓐ \(\frac{\sqrt{810{c}^{-3}{d}^{7}}}{\sqrt{1000{c}^{}{d}^{-1}}}\) ⓑ \(\frac{\sqrt[3]{24{a}^{7}{b}^{-1}}}{\sqrt[3]{-81{a}^{-2}{b}^{2}}}\)

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.

\neq

not equal
The two sides are different.

\pm

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

|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: Divide Radical Expressions

Divide radical expressions
Rationalize a one term denominator
Rationalize a two term denominator
no factors in the radicand have perfect powers of the index
no fractions in the radicand
no radicals in the denominator of a fraction
If
A radical expression is considered simplified if there are:

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.

Обиди се со себе.

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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