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Exercises Expressions, exponents and radicals

Example 7 from Algebra and Trigonometry, 1.3 Radicals and Rational Exponents

Subtracting Square Roots

Subtract $20 \sqrt{72 a^{3} b^{4} c} - 14 \sqrt{8 a^{3} b^{4} c} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Rewrite each term so they have equal radicands.

$\begin{matrix} 20 \sqrt{72 a^{3} b^{4} c} & = & 20 \sqrt{9} \sqrt{4} \sqrt{2} \sqrt{a} \sqrt{a^{2}} \sqrt{{(b^{2})}^{2}} \sqrt{c} \\ = & 20 (3) (2) | a | b^{2} \sqrt{2 a c} \\ = & 120 | a | b^{2} \sqrt{2 a c} \end{matrix}$
$\begin{matrix} 14 \sqrt{8 a^{3} b^{4} c} & = & 14 \sqrt{2} \sqrt{4} \sqrt{a} \sqrt{a^{2}} \sqrt{{(b^{2})}^{2}} \sqrt{c} \\ = & 14 (2) | a | b^{2} \sqrt{2 a c} \\ = & 28 | a | b^{2} \sqrt{2 a c} \end{matrix}$
Now the terms have the same radicand so we can subtract.

$120 | a | b^{2} \sqrt{2 a c} - 28 | a | b^{2} \sqrt{2 a c} = 92 | a | b^{2} \sqrt{2 a c}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Subtract $3 \sqrt{80 x} - 4 \sqrt{45 x} .$

How did it go?

PreviousAdding Square Roots Next Rationalizing a Denominator Containing a Single Term

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 1.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.