Question1 2 3 4
Which expression is equivalent to $\sqrt{\frac{55 x^{7} y^{6}}{11 x^{11} y^{8}}}$ ? Assume $x>0$ and $y>0$ . $\frac{x^{2} \sqrt{5}}{y}$ $\frac{y \sqrt{5}}{x^{2}}$ $\frac{\sqrt{5}}{x^{2} y}$ $\frac{x \sqrt{5}}{y}$

Studdy Solution

STEP 1

1. $x > 0$ and $y > 0$ , which allows us to simplify the expression without considering negative values or zero.
2. The expression can be simplified by reducing the fraction inside the square root.
3. The square root of a fraction can be expressed as the square root of the numerator divided by the square root of the denominator.

STEP 2

1. Simplify the fraction inside the square root.
2. Apply the square root to the simplified fraction.
3. Simplify the resulting expression.

STEP 3

Simplify the fraction inside the square root:
$\frac{55 x^{7} y^{6}}{11 x^{11} y^{8}}$
Divide both the numerator and the denominator by the common factor $11$ :
$\frac{5 x^{7} y^{6}}{x^{11} y^{8}}$
Now, simplify the powers of $x$ and $y$ :
$\frac{5}{x^{4} y^{2}}$

STEP 4

Apply the square root to the simplified fraction:
$\sqrt{\frac{5}{x^{4} y^{2}}} = \frac{\sqrt{5}}{\sqrt{x^{4} y^{2}}}$

STEP 5

Simplify the expression:
$\frac{\sqrt{5}}{\sqrt{x^{4}} \cdot \sqrt{y^{2}}}$
Since $\sqrt{x^{4}} = x^{2}$ and $\sqrt{y^{2}} = y$ , the expression becomes:
$\frac{\sqrt{5}}{x^{2} y}$
The expression equivalent to the given expression is:
$\boxed{\frac{\sqrt{5}}{x^{2} y}}$

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