Sure! Let's solve the given expression step-by-step:
[tex]\[ \sqrt{343 a^4} + a^2 \sqrt{175} - 3 \sqrt{7 a^4} \][/tex]
First, let's break down and simplify each term individually:
### Step 1: Simplify [tex]\(\sqrt{343 a^4}\)[/tex]
We can factor [tex]\(343\)[/tex] as [tex]\(343 = 7 \times 7 \times 7 = 7^3\)[/tex]. Therefore, we have:
[tex]\[ \sqrt{343 a^4} = \sqrt{7^3 a^4} \][/tex]
Recall that [tex]\(\sqrt{x^2} = x\)[/tex], we can write:
[tex]\[ \sqrt{7^3 a^4} = \sqrt{7^2 \cdot 7 \cdot a^4} = \sqrt{(7a^2)^2 \cdot 7} = 7a^2 \sqrt{7} \][/tex]
### Step 2: Simplify [tex]\(a^2 \sqrt{175}\)[/tex]
Next, let's consider [tex]\(175\)[/tex]. We can factor [tex]\(175\)[/tex] as [tex]\(175 = 25 \times 7 = 5^2 \times 7\)[/tex]. Hence,
[tex]\[ a^2 \sqrt{175} = a^2 \sqrt{5^2 \times 7} = a^2 \cdot 5 \sqrt{7} = 5a^2 \sqrt{7} \][/tex]
### Step 3: Simplify [tex]\(3 \sqrt{7 a^4}\)[/tex]
We know that:
[tex]\[ 7 a^4 = 7 \cdot (a^2)^2 \][/tex]
Therefore,
[tex]\[ 3 \sqrt{7 a^4} = 3 \sqrt{7 \cdot (a^2)^2} = 3 \sqrt{7} \cdot \sqrt{(a^2)^2} = 3 \sqrt{7} \cdot a^2 = 3a^2 \sqrt{7} \][/tex]
### Step 4: Combine all the terms
Now, let's combine all the simplified terms:
[tex]\[ \sqrt{343 a^4} + a^2 \sqrt{175} - 3 \sqrt{7 a^4} = 7a^2 \sqrt{7} + 5a^2 \sqrt{7} - 3a^2 \sqrt{7} \][/tex]
Combine the coefficients of [tex]\(a^2 \sqrt{7}\)[/tex]:
[tex]\[ (7 + 5 - 3) a^2 \sqrt{7} = 9a^2 \sqrt{7} \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{9a^2 \sqrt{7}} \][/tex]
