To simplify the given expression, we will perform the necessary algebraic operations step by step.
The given expression is:
[tex]\[ 27 \sqrt{x y^2} - 3 \sqrt{a^2 x y^2} + 4 y \sqrt{a^2 x} \][/tex]
### Step 1: Simplify [tex]\( 27 \sqrt{x y^2} \)[/tex]
We know that [tex]\( y^2 \)[/tex] is a perfect square, so:
[tex]\[ \sqrt{x y^2} = y \sqrt{x} \][/tex]
Thus,
[tex]\[ 27 \sqrt{x y^2} = 27 y \sqrt{x} \][/tex]
### Step 2: Simplify [tex]\( -3 \sqrt{a^2 x y^2} \)[/tex]
Here, [tex]\( a^2 \)[/tex] is also a perfect square, so:
[tex]\[ \sqrt{a^2 x y^2} = a y \sqrt{x} \][/tex]
Thus,
[tex]\[ -3 \sqrt{a^2 x y^2} = -3 a y \sqrt{x} \][/tex]
### Step 3: Simplify [tex]\( 4 y \sqrt{a^2 x} \)[/tex]
Again, since [tex]\( a^2 \)[/tex] is a perfect square:
[tex]\[ \sqrt{a^2 x} = a \sqrt{x} \][/tex]
Thus,
[tex]\[ 4 y \sqrt{a^2 x} = 4 y a \sqrt{x} \][/tex]
### Combining all simplified terms
Now we combine all the simplified expressions:
[tex]\[ 27 y \sqrt{x} - 3 a y \sqrt{x} + 4 y a \sqrt{x} \][/tex]
### Group similar terms
We can now combine like terms (those with [tex]\( y \sqrt{x} \)[/tex] and [tex]\( a y \sqrt{x} \)[/tex]):
[tex]\[ = 27 y \sqrt{x} - 3 a y \sqrt{x} + 4 a y \sqrt{x} \][/tex]
### Compute the coefficients
Firstly, for the terms with [tex]\( y \sqrt{x} \)[/tex]:
[tex]\[ 27 y \sqrt{x} = 27 y \sqrt{x} \][/tex]
For the terms with [tex]\( a y \sqrt{x} \)[/tex]:
[tex]\[ -3 a y \sqrt{x} + 4 a y \sqrt{x} = (4 a y - 3 a y) \sqrt{x} = a y \sqrt{x} \][/tex]
### Final simplified expression
Combining the results, we get:
[tex]\[ 27 y \sqrt{x} + a y \sqrt{x} \][/tex]
So the simplified expression is:
[tex]\[ 27 \sqrt{x y^2} - 3 \sqrt{a^2 x y^2} + 4 y \sqrt{a^2 x} = 4 y \sqrt{a^2 x} + 27 \sqrt{x y^2} - 3 \sqrt{a^2 x y^2} \][/tex]
