In the fourth week of Pre-Calculus 11 we continued the absolute value and radicals unit. This week we learned about addition and subtraction of radicals.
Addition and Subtraction of Radicals: To add and subtract radicals, the radicands have to be of the same value. Not only do the radicands have to be the same, but the index also has to be the same. The coefficients in front of the radical are added or subtracted, not the radicand.
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![\sqrt{23}+\sqrt[4]{23}](https://s0.wp.com/latex.php?latex=%5Csqrt%7B23%7D%2B%5Csqrt%5B4%5D%7B23%7D&bg=ffffff&fg=000000&s=0 "\sqrt{23}+\sqrt[4]{23}")
The example above cannot be added together because the index’s are not the same. Although, the radicals have the same radicand, the index of the radicals are different.
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The expression above could be added together because the radicand and the index are the same. I added the coefficients of the radical together and not the value inside the radical.
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In the example above, before adding/subtracting the radicals I simplified everything inside the radical. By simplifying the radical first, I was able to tell whether I could add/subtract the radicals together.
How to Multiply Radicals: To multiply radicals you have to multiply the coefficients by the coefficients and multiply the radicand by the radicand.
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It’s best to simplify the radical before multiplying, if possible. By simplifying the radical beforehand, it’ll be easier to multiply the radicals together because the numbers will be smaller.
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In the example above, I simplified the radicals to make the numbers more manageable to multiply. After simplifying the radicals I began to distribute inside the brackets.
How to Divide Radicals: When dividing a question that involves radicals you can only divide radicals by radicals and coefficients by coefficients.
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The example above cannot be simplified any further. Although 4 and 2 have a common factor of 2, they cannot be simplified because they are not in the same form.
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The most important rule for dividing radicals is that a radical cannot be left in the denominator of a fraction. If the denominator does contain a radical you have to rationalize the denominator.
Rationalize the Denominator: To rationalize the denominator with a radical you have to multiply the whole fraction by the denominator.
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When the denominator is a binomial containing a radical you have to multiply by the conjugate. To find the conjugate of a binomial you have to flip the sign in the middle of the binomial. The conjugate of 
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