Sure, let's simplify the expression [tex]\( 2 \sqrt{75 x y^4} \)[/tex] step-by-step.
### Step 1: Factor and simplify inside the square root
First, we recognize that 75 can be factored into its prime factors and then split the square root accordingly:
[tex]\[ 75 = 25 \times 3 \][/tex]
We can rewrite 75 as:
[tex]\[ \sqrt{75 x y^4} = \sqrt{25 \times 3 \times x \times y^4} \][/tex]
### Step 2: Simplify square roots of perfect squares
Next, we take the square root of the perfect squares inside the expression:
[tex]\[ \sqrt{25 \times 3 \times x \times y^4} = \sqrt{25} \times \sqrt{3} \times \sqrt{x} \times \sqrt{y^4} \][/tex]
We know:
[tex]\[ \sqrt{25} = 5 \][/tex]
[tex]\[ \sqrt{y^4} = y^2 \text{ (since } \sqrt{y^4} = (y^4)^{1/2} = y^{4/2} = y^2\text{) } \][/tex]
So, the expression becomes:
[tex]\[ \sqrt{25 \times 3 \times x \times y^4} = 5 \times \sqrt{3} \times \sqrt{x} \times y^2 \][/tex]
### Step 3: Combine the simplified parts
Now, we simplify the original expression by multiplying everything together:
[tex]\[ 2 \sqrt{75 x y^4} = 2 \times (5 \sqrt{3} \sqrt{x} y^2) \][/tex]
This simplifies to:
[tex]\[ 2 \times 5 \times \sqrt{3} \times \sqrt{x} \times y^2 \][/tex]
Calculating the multiplication:
[tex]\[ 2 \times 5 = 10 \][/tex]
So, the expression becomes:
[tex]\[ 10 \sqrt{3} \sqrt{x} y^2 \][/tex]
### Step 4: Express the final simplified form
Thus, the simplified form of [tex]\( 2 \sqrt{75 x y^4} \)[/tex] is:
[tex]\[ 10 \sqrt{3} \sqrt{x} y^2 \][/tex]
So, the final simplified expression is:
[tex]\[ 10 \sqrt{3} \sqrt{x y^4} \][/tex]
