Questions: x √(54 x^2 y^3)

Solution Steps

To simplify the expression \( x \sqrt{54 x^{2} y^{3}} \), we can first simplify the expression inside the square root. We can factor out perfect squares from the radicand and then simplify the square root. Finally, multiply the result by \( x \).

The given expression is \( x \sqrt{54 x^{2} y^{3}} \). We start by simplifying the expression inside the square root. The number 54 can be factored into \( 54 = 9 \times 6 \), where 9 is a perfect square. Thus, we can rewrite the expression as:

\[ x \sqrt{9 \times 6 \times x^{2} \times y^{3}} \]

Next, we extract the perfect squares from the square root. The perfect squares are \( 9 \) and \( x^{2} \). We can take the square root of these terms:

\[ x \times 3 \times x \times \sqrt{6 \times y^{3}} \]

This simplifies to:

\[ 3x^{2} \sqrt{6y^{3}} \]

Now, we simplify the remaining square root, \( \sqrt{6y^{3}} \). The term \( y^{3} \) can be rewritten as \( y^{2} \times y \), where \( y^{2} \) is a perfect square. Thus, we have:

\[ 3x^{2} \sqrt{6} \times \sqrt{y^{2} \times y} = 3x^{2} \sqrt{6} \times y \times \sqrt{y} \]

This simplifies to:

\[ 3x^{2}y \sqrt{6y} \]

Final Answer