University of Kwazulu-Natal

Multiplying Radical Expressions Chilimath

Bonisiwe Shabane
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multiplying radical expressions chilimath

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. 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. \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \) \( \newcommand{\dsum}{\displaystyle\sum\limits} \) \( \newcommand{\dint}{\displaystyle\int\limits} \)

\( \newcommand{\dlim}{\displaystyle\lim\limits} \) 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]

We multiply binomial expressions involving radicals by using the FOIL (First, Outer, Inner, Last) method. Example 1: Multiply each of the following Example 2: Multiply each of the following Exercise 1: Multiply each of the following A common way of dividing the radical expression is to have the denominator that contain no radicals. Dividing radical is based on rationalizing the denominator.

Rationalizing is the process of starting with a fraction containing a radical in its denominator and determining fraction with no radical in its denominator.

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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...

We Just Need To Tweak The Formula Above. But The

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. How do we multiply radical expressions together? For example, consider It is tempting to just multiply the two numbers together, but is that correct?

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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...

It Was Crucial In Our Process That The Exponents Were

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. \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \) \( \newcommand{\dsum}{\displaystyle\sum\limit...

\( \newcommand{\dlim}{\displaystyle\lim\limits} \) Answer: Use The Rule [latex] \sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}[/latex]

\( \newcommand{\dlim}{\displaystyle\lim\limits} \) 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{a...