Simplifying Square Roots by Factoring Out Perfect Squares and Combining Like Terms
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:
\( -26\sqrt{5} \)
