Let's carefully simplify the given mathematical expression step-by-step. The expression we need to simplify is:
[tex]\[ x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4} \][/tex]
### Step 1: Simplify each term inside the square root
#### Term 1: [tex]\( x \sqrt{5 x y^4} \)[/tex]
For the first term:
[tex]\[
x \sqrt{5 x y^4}
\][/tex]
Notice that [tex]\( y^4 \)[/tex] can be written as [tex]\( (y^2)^2 \)[/tex]. So, we rewrite the term inside the square root as:
[tex]\[
x \sqrt{5 x y^4} = x \sqrt{5 x (y^2)^2}
\][/tex]
We can take [tex]\( y^2 \)[/tex] out of the square root:
[tex]\[
= x \cdot y^2 \cdot \sqrt{5 x}
\][/tex]
So, the simplified first term is:
[tex]\[
x y^2 \sqrt{5 x}
\][/tex]
#### Term 2: [tex]\( \sqrt{405 x^3 y^4} \)[/tex]
For the second term:
[tex]\[
\sqrt{405 x^3 y^4}
\][/tex]
Rewrite the term inside the square root to see how we can simplify:
[tex]\[
\sqrt{405 x^3 y^4} = \sqrt{405 x^3 (y^2)^2}
\][/tex]
We can take [tex]\( y^2 \)[/tex] outside of the square root, and also factorize 405 as [tex]\( 9 \cdot 45 \)[/tex]:
[tex]\[
= y^2 \sqrt{405 x^3} = y^2 \sqrt{9 \cdot 45 x^3}
\][/tex]
Since [tex]\( \sqrt{9} = 3 \)[/tex]:
[tex]\[
= y^2 \cdot 3 \sqrt{45 x^3}
\][/tex]
Notice that inside the square root, [tex]\( 45x^3 \)[/tex] simplifies since [tex]\( x^3 \)[/tex] can be written as [tex]\( x \cdot x^2 \)[/tex]:
[tex]\[
= y^2 \cdot 3 \cdot x \sqrt{45 x} = y^2 \cdot 3 x \sqrt{45 x}
\][/tex]
#### Term 3: [tex]\( \sqrt{80 x^3 y^4} \)[/tex]
For the third term:
[tex]\[
\sqrt{80 x^3 y^4}
\][/tex]
Similar to the previous terms, rewrite the term inside the square root:
[tex]\[
\sqrt{80 x^3 y^4} = \sqrt{80 x^3 (y^2)^2}
\][/tex]
Take [tex]\( y^2 \)[/tex] outside of the square root and factorize 80 as [tex]\( 16 \cdot 5 \)[/tex]:
[tex]\[
= y^2 \sqrt{80 x^3} = y^2 \sqrt{16 \cdot 5 x^3}
\][/tex]
Since [tex]\( \sqrt{16} = 4 \)[/tex]:
[tex]\[
= y^2 \cdot 4 \sqrt{5 x^3}
\][/tex]
And simplify inside the square root:
[tex]\[
= y^2 \cdot 4 \cdot x \sqrt{5 x} = 4 x y^2 \sqrt{5 x}
\][/tex]
### Step 2: Combine the simplified expressions
Now, combine the terms:
[tex]\[
x y^2 \sqrt{5 x} + 3 x y^2 \sqrt{45 x} - 4 x y^2 \sqrt{5 x}
\][/tex]
Factor out the common term [tex]\( x y^2 \)[/tex]:
[tex]\[
x y^2 \left( \sqrt{5 x} + 3 \sqrt{45 x} - 4 \sqrt{5 x} \right)
\][/tex]
Simplify the inside of the parentheses:
[tex]\[
\sqrt{5 x} - 4 \sqrt{5 x} = -3 \sqrt{5 x}
\][/tex]
Thus, it now simplifies to:
[tex]\[
x y^2 \left( -3 \sqrt{5 x} + 3 \sqrt{45 x} \right)
\][/tex]
Use [tex]\( \sqrt{45 x} = \sqrt{9 \cdot 5 x} = 3 \sqrt{5 x} \)[/tex]:
[tex]\[
= x y^2 ( -3 \sqrt{5 x} + 3 \cdot 3 \sqrt{5 x}) = x y^2 (-3 \sqrt{5 x} + 9 \sqrt{5 x})
\][/tex]
Simplify again:
[tex]\[
= x y^2 (6 \sqrt{5 x})
\][/tex]
So, the fully simplified expression is:
[tex]\[
= 6 x y^2 \sqrt{5 x}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
6 x y^2 \sqrt{5 x}
\][/tex]
