University of Kwazulu-Natal

5 3 Multiplying Radical Expressions Intermediate Algebra

Bonisiwe Shabane
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5 3 multiplying radical expressions intermediate algebra

How do we multiply radical expressions together? For example, consider It is tempting to just multiply the two numbers together, but is that correct? The key is to use what we learned in the previous section. Remember that since radicals are really equivalent to rational exponents, every exponent property gives us a corresponding radical property! First recall our Power of a Product rule for exponents, which works for any rational exponent [latex]m[/latex] as long as both [latex]a^m[/latex] and [latex]b^m[/latex] are real numbers:

Now, evaluate the above expression using an exponent of [latex]\dfrac{1}{2}[/latex] to represent the square roots: [latex]\begin{align} &\quad\sqrt{2}\cdot \sqrt{18} \\ =&\quad 2^{1/2}\cdot18^{1/2} \\ =&\quad(2\cdot18)^{1/2} && \color{blue}{\textsf{Power of a Product used here}}\\ =&\quad(36)^{1/2} \\ =&\quad \sqrt{36} \\ =&\quad 6 \end{align}[/latex] So we have shown that you can indeed multiply radicals by multiplying the radicands. It was crucial in our process that the exponents were equal to be able to use the Power of a Product rule. Thus radical multiplication only works this way if the indices are the same. By the end of this section, you will be able to:

Before you get started, take this readiness quiz. Add: 3x2+9x−5−(x2−2x+3).3x2+9x−5−(x2−2x+3). If you missed this problem, review Example 5.5. Simplify: (2+a)(4−a).(2+a)(4−a). If you missed this problem, review Example 5.28. Simplify: (9−5y)2.(9−5y)2.

If you missed this problem, review Example 5.31. Answer: Use the rule [latex] \sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}[/latex] to multiply the radicands. [latex]\begin{array}{r}\sqrt{18\cdot 16}\\\sqrt{288}\end{array}[/latex] [latex] \sqrt{{{(12)}^{2}}\cdot 2}[/latex] [latex] \sqrt{{{(12)}^{2}}}\cdot \sqrt{2}[/latex] [latex]\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}[/latex]

In this lesson, we are only going to deal with square roots only which is a specific type of radical expression with an index of [latex]\color{red}2[/latex]. If you see a radical symbol without an index explicitly written, it is understood to have an index of [latex]\color{red}2[/latex]. Below are the basic rules in multiplying radical expressions. A radicand is a term inside the square root. We multiply radicals by multiplying their radicands together while keeping their product under the same radical symbol. What happens then if the radical expressions have numbers that are located outside?

We just need to tweak the formula above. But the key idea is that the product of numbers located outside the radical symbols remains outside as well. Let’s go over some examples to see how these two basic rules are applied. 1. Multiplication Property of Radicals 2. Expressions of the Form <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0012222012/814400/gray_6_5_2.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (9.0K)</a> 3.

Special Case Products 4. Multiplying Radicals with Different Indices Please read our Terms of Use and Privacy Notice before you explore our Web site. To report a technical problem with this Web site, please contact the Web Producer. \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\dsum}{\displaystyle\sum\limits} \) \( \newcommand{\dint}{\displaystyle\int\limits} \) \( \newcommand{\dlim}{\displaystyle\lim\limits} \) You can do more than just simplify radical expressions. You can multiply and divide them, too. The product raised to a power rule that we discussed previously will help us find products of radical expressions.

Recall the rule: For any numbers a and b and any integer x: [latex]{{(ab)}^{x}}={{a}^{x}}\cdot {{b}^{x}}[/latex] For any numbers a and b and any positive integer x: [latex]{{(ab)}^{\frac{1}{x}}}={{a}^{\frac{1}{x}}}\cdot {{b}^{\frac{1}{x}}}[/latex] For any numbers a and b and any positive integer x: [latex]\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex] The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that you cannot multiply a square root and a cube root using this rule.

The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands. Answer: Use the rule [latex] \sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}[/latex] to multiply the radicands. [latex]\begin{array}{r}\sqrt{18\cdot 16}\\\sqrt{288}\end{array}[/latex] [latex] \sqrt{{{(12)}^{2}}\cdot 2}[/latex] [latex] \sqrt{{{(12)}^{2}}}\cdot \sqrt{2}[/latex]

[latex]\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}[/latex]

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Before You Get Started, Take This Readiness Quiz. Add: 3x2+9x−5−(x2−2x+3).3x2+9x−5−(x2−2x+3).

Before you get started, take this readiness quiz. Add: 3x2+9x−5−(x2−2x+3).3x2+9x−5−(x2−2x+3). If you missed this problem, review Example 5.5. Simplify: (2+a)(4−a).(2+a)(4−a). If you missed this problem, review Example 5.28. Simplify: (9−5y)2.(9−5y)2.

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If you missed this problem, review Example 5.31. Answer: Use the rule [latex] \sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}[/latex] to multiply the radicands. [latex]\begin{array}{r}\sqrt{18\cdot 16}\\\sqrt{288}\end{array}[/latex] [latex] \sqrt{{{(12)}^{2}}\cdot 2}[/latex] [latex] \sqrt{{{(12)}^{2}}}\cdot \sqrt{2}[/latex] [latex]\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{arr...

In This Lesson, We Are Only Going To Deal With

In this lesson, we are only going to deal with square roots only which is a specific type of radical expression with an index of [latex]\color{red}2[/latex]. If you see a radical symbol without an index explicitly written, it is understood to have an index of [latex]\color{red}2[/latex]. Below are the basic rules in multiplying radical expressions. A radicand is a term inside the square root. We m...