Simplify. $-4 \sqrt{45}-\sqrt{20}$
Answer (2)
Simplify 45 to 3 5 .
Simplify 20 to 2 5 .
Substitute the simplified radicals into the expression: − 4 ( 3 5 ) − 2 5 .
Combine like terms to get the final answer: − 14 5 .
Explanation
Understanding the problem We are asked to simplify the expression $-4
\sqrt{45} - \sqrt{20}$. To do this, we need to simplify the square roots by factoring out perfect squares and then combine like terms.
Simplifying the first square root First, let's simplify 45 . We can factor 45 as 9 ⋅ 5 , where 9 is a perfect square. So, 45 = 9 ⋅ 5 = 9 ⋅ 5 = 3 5 .
Simplifying the second square root Next, let's simplify 20 . We can factor 20 as 4 ⋅ 5 , where 4 is a perfect square. So, 20 = 4 ⋅ 5 = 4 ⋅ 5 = 2 5 .
Substituting back into the expression Now, substitute the simplified square roots back into the original expression: − 4 45 − 20 = − 4 ( 3 5 ) − 2 5 .
Performing the multiplication Multiply: − 4 ( 3 5 ) = − 12 5 . So the expression becomes − 12 5 − 2 5 .
Combining like terms Now, we combine like terms: − 12 5 − 2 5 = ( − 12 − 2 ) 5 = − 14 5 .
Final Answer Therefore, the simplified expression is − 14 5 .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, when dealing with distances, areas, or volumes that involve square roots. For example, when calculating the length of the diagonal of a rectangle with sides of length 45 and 20 , you would need to simplify these radicals to find the total length using the Pythagorean theorem: d = ( 45 ) 2 + ( 20 ) 2 = 45 + 20 = 65 . Simplifying radicals also helps in approximating values and making calculations easier in practical applications.
The expression − 4 45 − 20 simplifies to − 14 5 by first simplifying the square roots, substituting them back into the expression, and combining like terms. The simplified forms of the square roots are 3 5 and 2 5 respectively. This results in an easy-to-read final expression.
;
