Certainly! Let's simplify the given expression step-by-step:
The expression we need to simplify is:
[tex]\[
\sqrt{80d} - \sqrt{20d}
\][/tex]
### Step 1: Factorize inside the square roots
We start by factorizing the expressions inside the square roots. Recall that [tex]\(80\)[/tex] and [tex]\(20\)[/tex] can both be broken down into prime factors. Specifically:
[tex]\[
80 = 16 \times 5
\][/tex]
[tex]\[
20 = 4 \times 5
\][/tex]
Substituting these into the square roots:
[tex]\[
\sqrt{80d} = \sqrt{16 \times 5 \times d}
\][/tex]
[tex]\[
\sqrt{20d} = \sqrt{4 \times 5 \times d}
\][/tex]
### Step 2: Simplify the square root expressions
We can use the property of square roots [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex] to simplify further:
[tex]\[
\sqrt{80d} = \sqrt{16} \cdot \sqrt{5} \cdot \sqrt{d}
\][/tex]
[tex]\[
\sqrt{20d} = \sqrt{4} \cdot \sqrt{5} \cdot \sqrt{d}
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex] and [tex]\(\sqrt{4} = 2\)[/tex], we substitute these values:
[tex]\[
\sqrt{80d} = 4 \cdot \sqrt{5} \cdot \sqrt{d}
\][/tex]
[tex]\[
\sqrt{20d} = 2 \cdot \sqrt{5} \cdot \sqrt{d}
\][/tex]
### Step 3: Combine the expressions
Now we substitute these simplified forms back into the original expression:
[tex]\[
\sqrt{80d} - \sqrt{20d} = \left( 4 \sqrt{5} \sqrt{d} \right) - \left( 2 \sqrt{5} \sqrt{d} \right)
\][/tex]
### Step 4: Factor out common terms
Notice that [tex]\(\sqrt{5} \sqrt{d}\)[/tex] is a common factor in both terms. We factor it out:
[tex]\[
4 \sqrt{5} \sqrt{d} - 2 \sqrt{5} \sqrt{d} = (4 - 2) \sqrt{5} \sqrt{d}
\][/tex]
### Step 5: Simplify the coefficients
Simplify the expression by performing the subtraction:
[tex]\[
(4 - 2) \sqrt{5} \sqrt{d} = 2 \sqrt{5} \sqrt{d}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
2 \sqrt{5} \sqrt{d}
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{2 \sqrt{5} \sqrt{d}}
\][/tex]
