Answer :
The simplified form of [tex]\(7\sqrt{28x^6} + 4\sqrt{7x^6}\)[/tex] is [tex]\(x(14 + 4x)\)[/tex].
To simplify the expression [tex]\(7\sqrt{28x^6} + 4\sqrt{7x^6}\)[/tex], you can first factor out any common factors from the radicands (the expressions inside the square roots).
Step 1: Factor out common factors from the radicands.
[tex]\(7\sqrt{28x^6}\)[/tex] can be simplified as follows:
[tex]\(7\sqrt{28x^6} = 7\sqrt{(4x^2)(7x^4)}\)[/tex]
Similarly, for [tex]\(4\sqrt{7x^6}\)[/tex], you can factor out the common factors:
[tex]\(4\sqrt{7x^6} = 4\sqrt{(1x^2)(7x^4)}\)[/tex]
Step 2: Simplify the square roots of the factored expressions.
Now, let's simplify each square root individually:
For [tex]\(7\sqrt{(4x^2)(7x^4)}\)[/tex], you can split it into two square roots:
[tex]\(7\sqrt{(4x^2)(7x^4)} = 7\sqrt{4x^2} \cdot \sqrt{7x^4}\)[/tex]
Simplify each square root separately:
[tex]\(7\sqrt{4x^2} = 7(2x) = 14x\)[/tex]
[tex]\(\sqrt{7x^4} = x^2\)[/tex]
Similarly, for [tex]\(4\sqrt{(1x^2)(7x^4)}\)[/tex], you can split it into two square roots:
[tex]\(4\sqrt{(1x^2)(7x^4)} = 4\sqrt{x^2} \cdot \sqrt{7x^4}\)[/tex]
Simplify each square root separately:
[tex]\(4\sqrt{x^2} = 4x\)[/tex]
[tex]\(\sqrt{7x^4} = x^2\)[/tex]
Step 3: Combine the simplified square roots.
Now that both square roots have been simplified, combine them back into a single expression:
[tex]\(14x + 4x^2\)[/tex]
Step 4: Factor out the common factor, which is x:
[tex]\(x(14 + 4x)\)[/tex]
So, the answer is [tex]\(x(14 + 4x)\)[/tex].
