Certainly! Let's solve the expression step by step.
Given:
[tex]\[
\sqrt{45w} - \sqrt{20w}
\][/tex]
First, we simplify each square root term individually.
Starting with [tex]\(\sqrt{45w}\)[/tex]:
1. Factor 45 into its prime factors:
[tex]\[
45 = 9 \times 5 = 3^2 \times 5
\][/tex]
2. Rewrite the expression:
[tex]\[
\sqrt{45w} = \sqrt{(3^2 \times 5 \times w)}
\][/tex]
3. Using the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we can simplify:
[tex]\[
\sqrt{(3^2 \times 5 \times w)} = \sqrt{3^2} \times \sqrt{5} \times \sqrt{w} = 3 \times \sqrt{5} \times \sqrt{w}
\][/tex]
Thus:
[tex]\[
\sqrt{45w} = 3 \sqrt{5} \sqrt{w}
\][/tex]
Next, we simplify [tex]\(\sqrt{20w}\)[/tex]:
1. Factor 20 into its prime factors:
[tex]\[
20 = 4 \times 5 = 2^2 \times 5
\][/tex]
2. Rewrite the expression:
[tex]\[
\sqrt{20w} = \sqrt{(2^2 \times 5 \times w)}
\][/tex]
3. Using the property of square roots:
[tex]\[
\sqrt{(2^2 \times 5 \times w)} = \sqrt{2^2} \times \sqrt{5} \times \sqrt{w} = 2 \times \sqrt{5} \times \sqrt{w}
\][/tex]
Thus:
[tex]\[
\sqrt{20w} = 2 \sqrt{5} \sqrt{w}
\][/tex]
Now, subtract the simplified terms:
[tex]\[
\sqrt{45w} - \sqrt{20w} = 3 \sqrt{5} \sqrt{w} - 2 \sqrt{5} \sqrt{w}
\][/tex]
Both terms have a common factor of [tex]\(\sqrt{5} \sqrt{w}\)[/tex]. Factor this common element out:
[tex]\[
\sqrt{5} \sqrt{w} (3 - 2) = \sqrt{5} \sqrt{w} \times 1 = \sqrt{5} \sqrt{w}
\][/tex]
Therefore:
[tex]\[
\sqrt{45w} - \sqrt{20w} = \sqrt{5w}
\][/tex]
