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Section 6.4 Working with Radicals

Subsection 1. Factor

To simplify a radical, we factor out the largest perfect square.

Subsubsection Examples

Example 6.47.

Find the missiing factor.

  1. \(60x^9 = 3x^3 \cdot~\) ?
  2. \(9x^3 - 3x^9 \cdot~\) ?
Solution
  1. We mentally divide \(60x^9\) by \(3x^3\) to find \(\dfrac{60x^9}{3x^3} = 20x^6\text{.}\) The missing factor is \(20x^6\text{.}\)
  2. We mentally divide each term by \(3x^3\) to find \(\dfrac{9x^3}{3x^3} = 3\) and \(\dfrac{3x^9}{3x^3}=x^6\text{.}\) The missing factor is \(3-x^6\text{.}\)
Example 6.48.

Factor out the largest perfect square.

  1. \(\displaystyle 108a^5b^2\)
  2. \(\displaystyle \dfrac{a^2+4a^4}{8}\)
Solution
  1. By trial and error, we find that 36 is the largest square that divides 108. From each power, we can factor out the power with the largest possible even exponent, namely \(a^4\) and \(b^2\text{.}\) Thus, we factor out \(36a^4b^2\) to find \(108a^5b^2 =36a^4b^2 \cdot 3a\text{.}\)
  2. The largest even power that divides into both \(a^2\) and \(a^4\) is \(a^2\text{,}\) so we factor \(a^2\) from the numerator:
    \begin{equation*} a^2+a^4=a^2(1+4a^2) \end{equation*}
    The largest perfect square that divides into the denominator is 4. Thus, we factor out \(\dfrac{a^2}{4}\) from the fraction to find
    \begin{equation*} \dfrac{a^2+4a^4}{8} = \dfrac{a^2}{4} \cdot \dfrac{1+4a^2}{2} \end{equation*}

Subsubsection Exercises

Find the missiing factor.

  1. \(16z^{16}+4z^6 = 4z^4 \cdot~\) ?
  2. \(\dfrac{20}{7}m^7 = 4m^6 \cdot~\) ?
Answer
  1. \(\displaystyle 4z^{12}+z^2\)
  2. \(\displaystyle \dfrac{5}{7}m\)

Factor out the largest perfect square.

  1. \(\displaystyle \dfrac{5k^5}{9n}\)
  2. \(\displaystyle 32a^{10}-48a^9\)
Answer
  1. \(\displaystyle \dfrac{k^4}{9} \cdot \dfrac{5k}{n}\)
  2. \(\displaystyle 16a^8(2a^2-3a)\)

Subsection 2. Apply properties of radicals

We have a Product Rule and a Quotient Rule for radicals.

Rules for Radicals.
  • \(\displaystyle \sqrt{ab} =\sqrt{a}~\sqrt{b}~~~~~~\text{if}~a,~b \ge 0\)
  • \(\displaystyle \sqrt{\dfrac{a}{b}} =\dfrac{\sqrt{a}}{\sqrt{b}}~~~~~~\text{if}~a \ge 0,~b \gt 0\)

Subsubsection Examples

Example 6.51.

Decide whether each statement is true or false. Then use a calculator to verify your answer.

  1. \(\displaystyle \sqrt{6} = \sqrt{2}~\sqrt{3}\)
  2. \(\displaystyle \sqrt{6} = \sqrt{2}+\sqrt{4}\)
Solution
  1. Yes: we can multiply (or divide) radicals together, if they have the same index. You can check that \(\sqrt{6} \approx 2.4495\text{,}\) and
    \begin{equation*} \sqrt{2}~\sqrt{3} \approx (1.4142)(1.7321) = 2.4495 \end{equation*}
    rounded to four decimal places.
  2. No: we cannot combine radicals with addition or subtraction. You can check that \(\sqrt{6} \approx 2.4495\text{,}\) but \(\sqrt{2}+\sqrt{4} \approx 1.4142+2=3.4142\text{.}\)
Example 6.52.

Find and correct the error in each calculation.

  1. \(\displaystyle \sqrt{36+64} \rightarrow 6+8\)
  2. \(\displaystyle \sqrt{3}+\sqrt{3} \rightarrow \sqrt{6}\)
Solution

  1. We cannot split radicals with addition or subtraction; we must follow the order of operations:

    \begin{equation*} = \sqrt{36+64} = \sqrt{100} = 10 \end{equation*}

  2. We cannot combine radicals with addition or subtraction. However, we can add like terms:

    \begin{equation*} \sqrt{3}+\sqrt{3} = 2\sqrt{3} \end{equation*}

Subsubsection Exercises

Decide whether each statement is true or false. Then use a calculator to verify your answer.

  1. \(\displaystyle \sqrt{16} = \sqrt{18} - \sqrt{2}\)
  2. \(\displaystyle \sqrt{8} = \dfrac{\sqrt{72}}{\sqrt{9}}\)
  3. \(\displaystyle \sqrt{5} + \sqrt{5} = \sqrt{10}\)
  4. \(\displaystyle \sqrt{2}~\sqrt{9} = \sqrt{18}\)
Answer
  1. False
  2. True
  3. False
  4. True

Find and correct the error in each calculation.

  1. \(\displaystyle \sqrt{25+5} \rightarrow 5+\sqrt{5}\)
  2. \(\displaystyle \sqrt{10}+\sqrt{15} \rightarrow 5\)
  3. \(\displaystyle \sqrt{9+x^2} \rightarrow 3+x\)
  4. \(\displaystyle \sqrt{a^2-b^2}\rightarrow a-b\)
Answer
  1. \(\displaystyle \sqrt{30}\)
  2. cannot be simplified
  3. cannot be simplified
  4. cannot be simplified

Subsection 3. Simplify radicals

We simplify square roots by factoring out any perfect squares.

Subsubsection Examples

Example 6.55.

Simplify \(~\sqrt{45}\)

Solution

The largest perfect square that divides evenly into 45 is 9, so we factor 45 as \(9 \cdot 5\text{.}\) We use the product rule to write

\begin{equation*} \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9}~\sqrt{5} \end{equation*}

Finally, we simplify to get

\begin{equation*} \sqrt{45} = \sqrt{9}~\sqrt{5} = 3\sqrt{5} \end{equation*}
Example 6.56.

Simplify \(~\sqrt{20x^2y^3}\)

Solution

The largest perfect square that divides 20 is 4. We write the radicand as the product of two factors, one containing the perfect square and the largest possible even powers of the variables. That is,

\begin{equation*} 20x^2y^3 = 4x^2y^2 \cdot 5y \end{equation*}

Then we write the radical as a product.

\begin{equation*} \sqrt{20x^2y^3} = \sqrt{4x^2y^2 \cdot 5y} = \sqrt{4x^2y^2}~\sqrt{5y} \end{equation*}

Finally, we simplify the first of the two factors to find

\begin{equation*} \sqrt{20x^2y^3}=\blert{\sqrt{4x^2y^2}}~\sqrt{5y} = \blert{2xy}~\sqrt{5y} \end{equation*}

Subsubsection Exercises

Simplify \(~\sqrt{75}\)

Answer
\(3\sqrt{5}\)

Simplify \(~\sqrt{72u^6v^9}\)

Answer
\(6u^3v^4\sqrt{2v}\)