Questions: Simplify the following expression completely, where x ≥ 0. x sqrt(5xy^4) + sqrt(405x^3y^4) - sqrt(80x^3y^4)

Transcript text: Simplify the following expression completely, where $\mathrm{x} \geq 0$. $x \sqrt{5 x y^{4}}+\sqrt{405 x^{3} y^{4}}-\sqrt{80 x^{3} y^{4}}$

Solution

Solution Steps

Step 1: Expand the Expression

Start with the original expression: \[ x \sqrt{5 x y^{4}} + \sqrt{405 x^{3} y^{4}} - \sqrt{80 x^{3} y^{4}} \]

Step 2: Simplify Each Term

Simplify each term separately:

For $ x \sqrt{5 x y^{4}} $: \[ x \sqrt{5} \sqrt{x} \sqrt{y^{4}} = x y^{2} \sqrt{5 x} \]
For $ \sqrt{405 x^{3} y^{4}} $: \[ \sqrt{405} \sqrt{x^{3}} \sqrt{y^{4}} = 9 \sqrt{5} x^{3/2} y^{2} \]
For $ \sqrt{80 x^{3} y^{4}} $: \[ \sqrt{80} \sqrt{x^{3}} \sqrt{y^{4}} = 4 \sqrt{5} x^{3/2} y^{2} \]

Step 3: Combine Like Terms

Now combine the simplified terms: \[ x y^{2} \sqrt{5 x} + 9 \sqrt{5} x^{3/2} y^{2} - 4 \sqrt{5} x^{3/2} y^{2} \] This simplifies to: \[ x y^{2} \sqrt{5 x} + (9 - 4) \sqrt{5} x^{3/2} y^{2} = x y^{2} \sqrt{5 x} + 5 \sqrt{5} x^{3/2} y^{2} \]

Step 4: Factor Out Common Terms

Factor out the common terms $ y^{2} \sqrt{5} x^{3/2} $: \[ = 6 \sqrt{5} x^{3/2} y^{2} \]

Thus, the final simplified expression is: \[ 6 \sqrt{5} x^{3/2} y^{2} \]