ORCCA Radical Expression Operations

Section 13.3 Radical Expression Operations

In Section 9.1, we learned square root properties. In this section, we will extend those concepts into radicals with higher roots and learn how to handle radical expressions with variables.

Subsection 13.3.1 Simplifying Radical Expressions

In Section 9.1, we learned how to simplify radical expressions like \(\sqrt{12}=2\sqrt{3}\text{.}\) In this subsection, we will use the same concept to simplify radical expressions like \(\sqrt[3]{16x^3}\text{.}\)

Recall that we learned this property for square roots:

\begin{equation*} \sqrt{x^2}=x\quad\text{where }x\ge0 \end{equation*}

The condition \(x\ge0\) must be true because this property doesn't hold if \(x\lt0\text{.}\) For example, if \(x=-1\text{:}\) \(\sqrt{(-1)^2}=\sqrt{1}=1\text{,}\) where \(\sqrt{(-1)^2}\ne-1\text{.}\) Therefore, we will use a different way to write this property so that it will be true for all values of \(x\text{:}\)

\begin{equation*} \sqrt{x^2}=\abs{x} \end{equation*}

This new formula will take care of negative numbers, like in \(\sqrt{(-1)^2}=\abs{-1}=1\text{.}\)

Recall that we have also learned that

\begin{equation*} \sqrt[n]{x}=x^{\frac{1}{n}} \end{equation*}

and it's easy to see

\begin{align*} \sqrt{x^2}\amp=(x^2)^{\frac{1}{2}}\\ \amp=x^{2\cdot\frac{1}{2}}\\ \amp=\abs{x} \end{align*}

Similarly, we can show the following:

\begin{align*} \sqrt{x^2}\amp=\abs{x}\\ \sqrt[4]{x^4}\amp=\abs{x}\\ \sqrt[6]{x^6}\amp=\abs{x}\\ \quad\ldots\\ \sqrt[n]{x^n}\amp=\abs{x}\amp\text{where }n\text{ is even} \end{align*}

However, for \(\sqrt[3]{x^3}=x\text{,}\) there is no need for the absolute value symbol. For example, \(\sqrt[3]{1^3}=1\) and \(\sqrt[3]{(-1)^3}=-1\text{.}\) In fact, it is true for all values of \(x\text{,}\) even negative values, that \(\sqrt[3]{x^3}=x\text{.}\) The same is true for other odd-number-degree roots.

In summary, we have:

\begin{align*} \sqrt[m]{x^m}\amp=x\amp\text{where }m\text{ is odd}\\ \sqrt[n]{x^n}\amp=\abs{x}\amp\text{where }n\text{ is even} \end{align*}

Let's look at a few examples on simplifying radicals.

Example 13.3.1.

Simplify \(\sqrt{y^2+2y+1}\text{.}\)

Explanation

\begin{align*} \sqrt{y^2+2y+1}\amp=\sqrt{(y+1)^2}\\ \amp=\abs{y+1} \end{align*}

We used the property \(\sqrt[n]{x^n}=\abs{x}\text{ where }n\text{ is even}\text{.}\)

Example 13.3.2.

Simplify \(\sqrt{z^4}\text{.}\)

Explanation

\begin{align*} \sqrt{z^4}\amp=\sqrt{\left(z^2\right)^2}\\ \amp=\abs{z^2}\\ \amp=z^2 \end{align*}

In the last example, we had an additional step to simplify, rather than leave our answer as \(\sqrt{z^4}=\abs{z^2}\text{.}\) Since \(z^2\) is always positive or zero, the absolute value symbols are redundant. This is always something to consider when simplifying these expressions.

Example 13.3.3.

Simplify \(\sqrt{72m^3n^5}\text{.}\) Assume all variables are positive.

Explanation

\begin{align*} \sqrt{72m^3n^5}\amp=\sqrt{36\cdot 2\cdot m^2\cdot m\cdot n^4\cdot n}\\ \amp=\sqrt{6^2\cdot 2\cdot m^2\cdot m\cdot \left(n^2\right)^2\cdot n}\\ \amp=6mn^2\sqrt{2mn} \end{align*}

Since all variables are assumed positive, there is no need to use the absolute value symbols.

Example 13.3.4.

Simplify \(\sqrt[3]{16z^4}\text{.}\) Assume all variables are positive.

Explanation

\begin{align*} \sqrt[3]{16z^4}\amp=\sqrt[3]{8\cdot 2\cdot z^3\cdot z}\\ \amp=\sqrt[3]{2^3\cdot 2\cdot z^3\cdot z}\\ \amp=2z\sqrt[3]{2z} \end{align*}

We used the property \(\sqrt[m]{x^m}=x\text{ where }m\text{ is odd}\text{.}\)

Example 13.3.5.

Simplify \(\sqrt[3]{54x^3y^5}\text{.}\)

Explanation

\begin{align*} \sqrt[3]{54x^3y^5}\amp=\sqrt[3]{27\cdot 2\cdot x^3\cdot y^3\cdot y^2}\\ \amp=\sqrt[3]{3^3\cdot 2\cdot x^3\cdot y^3\cdot y^2}\\ \amp=3xy\sqrt[3]{2y^2} \end{align*}

Subsection 13.3.2 Radical Operations

In Section 9.1, we learned how to add/subtract/multiply/divide square root expressions. We will now extend those concepts and simplify more complicated radical expressions.

In Section 9.1, we learned the properties:

\begin{align*} \sqrt{x\cdot y}\amp=\sqrt{x}\cdot\sqrt{y}\\ \sqrt{\frac{x}{y}}\amp=\frac{\sqrt{x}}{\sqrt{y}}, \text{ where }y\ne0 \end{align*}

Those properties also work for other radicals:

\begin{align*} \sqrt[n]{x\cdot y}\amp=\sqrt[n]{x}\cdot\sqrt[n]{y}\\ \sqrt[n]{\frac{x}{y}}\amp=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}, \text{ where }y\ne0 \end{align*}

The next few examples will use those properties.

Example 13.3.6.

Do multiplication: \(\sqrt[3]{4x^2y}\cdot 2\cdot \sqrt[3]{4xy^2}\)

Explanation

\begin{align*} \sqrt[3]{4x^2y}\cdot 2\cdot \sqrt[3]{4xy^2}\amp=2\cdot\sqrt[3]{4x^2y\cdot 4xy^2}\\ \amp=2\cdot \sqrt[3]{8\cdot 2\cdot x^3\cdot y^3}\\ \amp=2\cdot \sqrt[3]{2^3\cdot 2\cdot x^3\cdot y^3}\\ \amp=2\cdot 2xy\cdot \sqrt[3]{2}\\ \amp=4xy\sqrt[3]{2} \end{align*}

Example 13.3.7.

Do multiplication: \(\sqrt{2p^2-p-1}\cdot\sqrt{p^2-1}\)

Explanation

\begin{align*} \sqrt{2p^2-p-1}\cdot\sqrt{p^2-1}\amp=\sqrt{(2p+1)(p-1)}\cdot\sqrt{(p+1)(p-1)}\\ \amp=\sqrt{(2p+1)(p-1)(p+1)(p-1)}\\ \amp=\sqrt{(2p+1)(p-1)^2(p+1)}\\ \amp=\abs{p-1}\sqrt{(2p+1)(p+1)} \end{align*}

Example 13.3.8.

Do division: \(\frac{\sqrt[4]{32m^6n^{10}}}{\sqrt[4]{162m^{10}n}}\text{.}\) Assume all variables are positive.

Explanation

\begin{align*} \frac{\sqrt[4]{32m^6n^{10}}}{\sqrt[4]{162m^{10}n}}\amp=\sqrt[4]{\frac{32m^6n^{10}}{162m^{10}n}}\\ \amp=\sqrt[4]{\frac{16n^9}{81m^4}}\\ \amp=\sqrt[4]{\frac{2^4n^8n}{3^4m^4}}\\ \amp=\frac{2n^2\sqrt[4]{n}}{3m} \end{align*}

Notice that in this process we simplified \(\sqrt[4]{n^8}\) to \(n^2\text{.}\) This can be explained in a couple of different ways. First, we can rewrite \(\sqrt[4]{n^8}\) as \(\sqrt[4]{\left(n^2\right)^4}=n^2\) and use the property that \(\sqrt[m]{x^m}=x\) when all variables are positive. A second explanation would be to use the rational exponent rule by rewriting \(\sqrt[4]{n^8}\) as \(n^{\frac{8}{4}}=n^2\text{.}\) Either way, we've shown that \(\sqrt[4]{n^8}=n^2\text{.}\)

When we add/subtract radical expressions, remember we can only combine like terms.

Example 13.3.9.

Simplify \(\sqrt{8t}+\sqrt[3]{16t}+\sqrt{18t}\text{.}\) Assume all variables are positive.

Explanation

\begin{align*} \sqrt{8t}+\sqrt[3]{16t}+\sqrt{18t}\amp=\sqrt{4\cdot 2t}+\sqrt[3]{8\cdot 2t}+\sqrt{9\cdot 2t}\\ \amp=2\sqrt{2t}+2\cdot\sqrt[3]{2t}+3\sqrt{2t}\\ \amp=5\sqrt{2t}+2\sqrt[3]{2t} \end{align*}

Note that \(\sqrt{2t}\) and \(\sqrt[3]{2t}\) are not like terms.

Example 13.3.10.

Simplify \(\sqrt[3]{375n^{12}x^9}-\sqrt[3]{192n^{12}x^9}\text{.}\) Assume all variables are positive.

Explanation

\begin{align*} \sqrt[3]{375n^{12}x^9}-\sqrt[3]{192n^{12}x^9}\amp=\sqrt[3]{125\cdot 3 n^{12}x^9}-\sqrt[3]{64\cdot 3n^{12}x^9}\\ \amp=\sqrt[3]{5^3\cdot 3 \left(n^4\right)^3\left(x^3\right)^3}-\sqrt[3]{4^3\cdot 3\left(n^4\right)^3\left(x^3\right)^3}\\ \amp=5n^4x^3\sqrt[3]{3}-4n^4x^3\sqrt[3]{3}\\ \amp=n^4x^3\sqrt[3]{3} \end{align*}

When we add/subtract fractions, don't forget to find common denominators first.

Example 13.3.11.

Simplify \(\sqrt{\frac{3}{4}}+\sqrt{12}\text{.}\)

Explanation

\begin{align*} \sqrt{\frac{3}{4}}+\sqrt{12}\amp=\frac{\sqrt{3}}{\sqrt{4}}+\sqrt{4\cdot3}\\ \amp=\frac{\sqrt{3}}{2}+2\sqrt{3}\\ \amp=\frac{\sqrt{3}}{2}+\frac{2\sqrt{3}}{1}\\ \amp=\frac{\sqrt{3}}{2}+\frac{\multiplyleft{2}2\sqrt{3}}{\multiplyleft{2}1}\\ \amp=\frac{\sqrt{3}}{2}+\frac{4\sqrt{3}}{2}\\ \amp=\frac{\sqrt{3}+4\sqrt{3}}{2}\\ \amp=\frac{5\sqrt{3}}{2} \end{align*}

Exercises 13.3.3 Exercises

Simplifying Radicals

1.

Simplify the radical expression or state that it is not a real number.

\(\displaystyle{ {\sqrt{66}} = }\)

2.

Simplify the radical expression or state that it is not a real number.

\(\displaystyle{ {\sqrt{52}} = }\)

3.

Simplify the radical expression or state that it is not a real number.

\(\displaystyle{ {\sqrt{392}} = }\)

4.

Simplify \(\displaystyle{ \sqrt[3]{108} }\text{.}\)

\(\displaystyle{ \sqrt[3]{108} =}\) \(\displaystyle{\cdot \sqrt[3]{(}}\) \(\displaystyle{)}\)

5.

Simplify \(\displaystyle{ \sqrt[4]{324} }\text{.}\)

\(\displaystyle{ \sqrt[4]{324} =}\) \(\displaystyle{\cdot \sqrt[4]{(}}\) \(\displaystyle{)}\)

6.

Simplify the following expressions. Do not assume the variables take only positive values.

\(\displaystyle \sqrt[3]{y^3}\)
\(\displaystyle \sqrt{y^2}\)

7.

Simplify the following expression. Do not assume the variables take only positive values.

\(\sqrt{9t^2-6t+1}\)

8.

Simplify the following expression. Do not assume the variables take only positive values.

\(\sqrt[3]{64(r+4)^3}\)

Simplify the following expressions. Assume all variables are positive.

9.

\(\sqrt{8m^8}\)

10.

\(\sqrt{280n^4}\)

11.

\(\sqrt{60a^9m^7}\)

12.

\(\sqrt[3]{-280b^2}\)

13.

\(\sqrt[3]{360b^6c^6}\)

14.

\(\sqrt[4]{80r^{11}x^9}\)

Multiplying Radicals

15.

Simplify the expression.

\(4\sqrt{7} \cdot 6\sqrt{2}=\)

16.

Simplify the expression.

\(5\sqrt{13} \cdot 4\sqrt{{25}} =\)

17.

Simplify the expression.

\(\displaystyle{4\sqrt{6} \cdot 5\sqrt{12}=}\)

18.

Simplify the expression.

\(\displaystyle{ {\sqrt{3}} \cdot {7\sqrt{27}} = }\)

Do the multiplication and simplify the result. Assume all variables are positive.

19.

\(\sqrt{n^3}\cdot\sqrt{n^5}\)

20.

\(\sqrt{3n}\cdot 8\sqrt{27n^3}\)

21.

\(\sqrt{27b^2x^3}\cdot\sqrt{2x^4}\)

22.

Multiply and simplify the result. Do not assume the variables take only positive values.

\(\sqrt{c^2+7c+12}\cdot\sqrt{c^2-c-20}\)

23.

Simplify \(\sqrt[3]{9}\cdot\sqrt[3]{45}\text{.}\)

24.

Simplify \(\sqrt[4]{20}\cdot\sqrt[4]{8}\text{.}\)

25.

Simplify \(\sqrt[3]{5z^4}\cdot\sqrt[3]{80}\text{.}\) Assume all variables are positive.

26.

Simplify \(\sqrt[4]{12r^3z^3}\cdot\sqrt[4]{36rz^3}\text{.}\) Assume all variables are positive.

27.

Simplify the radical expression or state that it is not a real number.

\(\displaystyle{ \frac{{\sqrt{108}}}{{\sqrt{3}}} =}\)

28.

Simplify the radical expression or state that it is not a real number.

\(\displaystyle{ \frac{{\sqrt{6}}}{{\sqrt{216}}} =}\)

29.

Simplify the expression.

\(\displaystyle{ \sqrt{\frac{3}{2}} \cdot \sqrt{\frac{5}{2}} =}\)

30.

Simplify the expression.

\(\displaystyle{ {\sqrt{\frac{14}{17}}} \cdot {\sqrt{\frac{2}{17}}} =}\)

31.

Simplify.

\(\sqrt[3]{\frac{32}{125}}\)

32.

Simplify.

\(\frac{\sqrt[3]{3}}{\sqrt[3]{-375}}\)

Simplify. Assume all variables are positive.

33.

\(\sqrt[3]{\frac{-125x^7}{z^3}}\)

34.

\(\frac{\sqrt[3]{zm^{18}}}{\sqrt[3]{z^{13}m^3}}\)

35.

Simplify the expression \(\displaystyle{ {\sqrt{\frac{8a^{7}}{5}}} \cdot {\sqrt{\frac{245a}{8}}} }\text{.}\) Assume all variables are positive.

\(\displaystyle{ {\sqrt{\frac{8a^{7}}{5}}} \cdot {\sqrt{\frac{245a}{8}}} =}\)

36.

\(\sqrt[3]{\frac{-6m^4}{7}}\cdot\sqrt[3]{\frac{875}{6m^{13}}}\)

Mixed Operations of Radicals

37.

Simplify the following expressions, if possible. Assume all variables are positive.

\(\displaystyle 3+5\sqrt{a}\)
\(\displaystyle 3\cdot 5\sqrt{a}\)

38.

Simplify the following expressions, if possible. Assume all variables are positive.

\(\displaystyle \sqrt{b}+8\sqrt{b}\)
\(\displaystyle \sqrt{b}\cdot 8\sqrt{b}\)

39.

Simplify, if possible.

\(9\sqrt{5}+6\sqrt[3]{5}\)

40.

Simplify the expression.

\(\displaystyle{{8\sqrt{x}} + {10\sqrt{x}} =}\)

41.

Simplify the expression.

\(\displaystyle{{15\sqrt{3}} - {16\sqrt{3}} =}\)

42.

Simplify the expression.

\(\displaystyle{{16\sqrt{15}} - {18\sqrt{15}} + {17\sqrt{15}} =}\)

43.

Simplify the expression.

\(\displaystyle{{52\sqrt{11r}} - {36\sqrt{11r}} + {20\sqrt{11r}} =}\)

44.

Simplify the expression.

\(\displaystyle{{\sqrt{28}} + {\sqrt{112}} =}\)

45.

Simplify the expression.

\(\displaystyle{{\sqrt{28}} + {\sqrt{252}} + {\sqrt{80}} + {\sqrt{125}} =}\)

46.

Simplify the expression.

\(\displaystyle{{\sqrt{32}} - {\sqrt{50}} =}\)

47.

Simplify the expression.

\(\displaystyle{{\sqrt{50}} - {\sqrt{8}} - {\sqrt{125}} - {\sqrt{125}} =}\)

Simplify. Assume all variables are positive.

48.

\(\sqrt{216b^7n^4}+\sqrt{54b^7n^4}\)

49.

\(\sqrt{250}+\sqrt{90c^2}\)

50.

\(\sqrt[3]{250}+\sqrt[3]{-54}\)

51.

\(\sqrt[3]{243m^5y^5}+\sqrt[3]{72m^5y^5}\)

52.

\(\sqrt[4]{567t^8}+\sqrt[4]{112b^8}\)

53.

Simplify.

\(\sqrt[4]{144}+\sqrt[4]{729}\)

54.

Simplify.

\(\frac{3\sqrt{10}}{2}+\frac{3\sqrt{10}}{4}\)

Simplify. Don't forget to rationalize the denominator.

55.

\(\sqrt{\frac{3}{5}}+\sqrt{15}\)

56.

\(\sqrt[3]{\frac{5}{2}}+\sqrt[3]{\frac{-4}{25}}\)

57.

\(\sqrt[3]{\frac{7}{2}}+\sqrt[3]{\frac{-4}{49}}\)

Simplify. Don't forget to rationalize the denominator. Assume all variables are positive

58.

\(\sqrt{a^3b}-\sqrt{\frac{b}{a}}\)

59.

\(\sqrt{\frac{7}{5b^2}}-\sqrt{\frac{5}{7}}\)

60.

\(\sqrt{\frac{5}{3y^2}}-\sqrt{\frac{3}{5}}\)