Questions: Rationalize the denominator: sqrt(3 x^5) / sqrt(7 x^11) = [square] / [square] To enter a number like 5 sqrt(7)

Solution Steps

To rationalize the denominator, we need to eliminate the square root in the denominator. This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator. Simplify the resulting expression.

Given the expression: \[ \frac{\sqrt{3 x^{5}}}{\sqrt{7 x^{11}}} \]

We can rewrite the numerator and the denominator as: \[ \frac{\sqrt{3} \cdot \sqrt{x^5}}{\sqrt{7} \cdot \sqrt{x^{11}}} \]

Combine the radicals in the numerator and the denominator: \[ \frac{\sqrt{3 x^5}}{\sqrt{7 x^{11}}} = \frac{\sqrt{3} \cdot x^{5/2}}{\sqrt{7} \cdot x^{11/2}} \]

Simplify the fraction by dividing the exponents of \(x\): \[ \frac{\sqrt{3} \cdot x^{5/2}}{\sqrt{7} \cdot x^{11/2}} = \frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{x^{5/2}}{x^{11/2}} = \frac{\sqrt{3}}{\sqrt{7}} \cdot x^{(5/2 - 11/2)} = \frac{\sqrt{3}}{\sqrt{7}} \cdot x^{-3} \]

To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{7}\): \[ \frac{\sqrt{3}}{\sqrt{7}} \cdot x^{-3} = \frac{\sqrt{3} \cdot \sqrt{7}}{7} \cdot x^{-3} = \frac{\sqrt{21}}{7} \cdot x^{-3} \]

Final Answer