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Simplify as much as possible.
[tex]\[ y \sqrt{28 x^3} + 6x \sqrt{7xy^2} \][/tex]
Assume that all variables represent positive real numbers.
[tex]\[ y \sqrt{28 x^3} + 6 x \sqrt{7 x y^2} \][/tex]
we need to break it down into manageable parts.
### Step 1: Simplify Each Term Individually
#### Term 1: [tex]\( y \sqrt{28 x^3} \)[/tex]
1. Recognize that we can break the expression inside the square root into prime factors: [tex]\[
\sqrt{28 x^3} = \sqrt{4 \cdot 7 \cdot x^3}
\][/tex] 2. Simplify the square root of each factor where possible: [tex]\[
\sqrt{4 \cdot 7 \cdot x^3} = \sqrt{4} \cdot \sqrt{7} \cdot \sqrt{x^3} = 2 \sqrt{7} \cdot x^{3/2}
\][/tex] 3. Substitute this back into the original term: [tex]\[
y \sqrt{28 x^3} = y \cdot 2 \sqrt{7} \cdot x^{3/2} = 2 y \sqrt{7} \cdot x^{3/2}
\][/tex]
#### Term 2: [tex]\( 6 x \sqrt{7 x y^2} \)[/tex]
1. Again break down the expression inside the square root: [tex]\[
\sqrt{7 x y^2} = \sqrt{7} \cdot \sqrt{x} \cdot \sqrt{y^2}
\][/tex] 2. Simplify the square root of each factor where possible: [tex]\[
\sqrt{7 x y^2} = \sqrt{7} \cdot \sqrt{x} \cdot y
\][/tex] 3. Substitute this back into the original term: [tex]\[
6 x \sqrt{7 x y^2} = 6 x \cdot \sqrt{7} \cdot \sqrt{x} \cdot y = 6 y \sqrt{7} \cdot x \cdot \sqrt{x} = 6 y \sqrt{7} \cdot x^{3/2}
\][/tex]
### Step 2: Combine the Simplified Terms
Now combining the two simplified terms together:
1. From Term 1: [tex]\[
2 y \sqrt{7} \cdot x^{3/2}
\][/tex]
2. From Term 2: [tex]\[
6 y \sqrt{7} \cdot x^{3/2}
\][/tex]
Combine them: [tex]\[
2 y \sqrt{7} \cdot x^{3/2} + 6 y \sqrt{7} \cdot x^{3/2}
\][/tex]
### Step 3: Factor Out Common Factors
1. Both terms have a common factor of [tex]\( y \sqrt{7} x^{3/2} \)[/tex]: [tex]\[
(2 + 6) y \sqrt{7} \cdot x^{3/2}
\][/tex] 2. Simplify inside the parentheses: [tex]\[
(2 + 6) y \sqrt{7} \cdot x^{3/2} = 8 y \sqrt{7} \cdot x^{3/2}
\][/tex]
Therefore, the simplified form of the given expression is:
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