Question - Simplifying Square Roots by Factoring Out Perfect Squares and Combining Like Terms
Solution:
To solve the expression given in the image, you need to simplify the square roots by factoring out perfect squares and then combine like terms.The expression is:$$ 5\sqrt{20} - 12\sqrt{45} $$First, let's simplify $$ \sqrt{20} $$ and $$ \sqrt{45} $$.For $$ \sqrt{20} $$, we look for the largest perfect square that is a factor of 20, which is 4. So we can write 20 as $$ 4 \times 5 $$, then take the square root of each factor:$$ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5} $$Now for $$ \sqrt{45} $$, the largest perfect square factor is 9. So we can write 45 as $$ 9 \times 5 $$, then take the square root of each factor:$$ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \sqrt{5} $$Now, we replace the square roots in the original expression with their simplified forms:$$ 5\cdot2\sqrt{5} - 12\cdot3\sqrt{5} $$This simplifies to:$$ 10\sqrt{5} - 36\sqrt{5} $$Now we can combine like terms:$$ (10 - 36)\sqrt{5} $$$$ -26\sqrt{5} $$So, the simplified form of the original expression is:$$ -26\sqrt{5} $$
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