Simplify ( square root of 25x)/( square root of 5y)

Solution:

Step:1

Merge the radicals $\sqrt{25 x}$ and $\sqrt{5 y}$ into a single square root expression: $\sqrt{\frac{25 x}{5 y}}$.

Step:2

Simplify the fraction $\frac{25 x}{5 y}$ by removing common factors.

Step:2.1

Extract the factor of $5$ from $25 x$: $\sqrt{\frac{5(5 x)}{5 y}}$.

Step:2.2

Extract the factor of $5$ from $5 y$: $\sqrt{\frac{5(5 x)}{5(y)}}$.

Step:2.3

Eliminate the common factor of $5$: $\sqrt{\frac{\cancel{5}(5 x)}{\cancel{5} y}}$.

Step:2.4

Reformulate the simplified expression: $\sqrt{\frac{5 x}{y}}$.

Step:3

Express $\sqrt{\frac{5 x}{y}}$ as a quotient of radicals: $\frac{\sqrt{5 x}}{\sqrt{y}}$.

Step:4

Multiply $\frac{\sqrt{5 x}}{\sqrt{y}}$ by the fraction $\frac{\sqrt{y}}{\sqrt{y}}$ to rationalize the denominator.

Step:5

Simplify the denominator by combining terms.

Step:5.1

Multiply the numerators and denominators involving square roots: $\frac{\sqrt{5 x} \sqrt{y}}{\sqrt{y} \sqrt{y}}$.

Step:5.2

Represent $\sqrt{y}$ as a power: $\frac{\sqrt{5 x} \sqrt{y}}{(\sqrt{y})^{1} \sqrt{y}}$.

Step:5.3

Repeat the representation of $\sqrt{y}$ as a power: $\frac{\sqrt{5 x} \sqrt{y}}{(\sqrt{y})^{1} (\sqrt{y})^{1}}$.

Step:5.4

Apply the exponent multiplication rule: $\frac{\sqrt{5 x} \sqrt{y}}{(\sqrt{y})^{1 + 1}}$.

Step:5.5

Sum the exponents: $\frac{\sqrt{5 x} \sqrt{y}}{(\sqrt{y})^{2}}$.

Step:5.6

Convert the square of a square root back to the original value.

Step:5.6.1

Use the radical to exponent conversion: $\frac{\sqrt{5 x} \sqrt{y}}{((y^{\frac{1}{2}}))^{2}}$.

Step:5.6.2

Apply the power of a power rule: $\frac{\sqrt{5 x} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$.

Step:5.6.3

Multiply the exponents: $\frac{\sqrt{5 x} \sqrt{y}}{y^{\frac{2}{2}}}$.

Step:5.6.4

Simplify the exponent by cancelling out common factors.

Step:5.6.4.1

Cancel out the common factors in the exponent: $\frac{\sqrt{5 x} \sqrt{y}}{y^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step:5.6.4.2

Rephrase the expression: $\frac{\sqrt{5 x} \sqrt{y}}{y^{1}}$.

Step:5.6.5

Final simplification: $\frac{\sqrt{5 x} \sqrt{y}}{y}$.

Step:6

Combine the radicals using the product rule: $\frac{\sqrt{5 x y}}{y}$.

Knowledge Notes:

• Radicals: A radical expression includes a root symbol and represents the root of a number or expression. The square root symbol $\sqrt{}$ is used for square roots.

• Combining Radicals: Radicals with the same index and radicand (the number or expression inside the radical) can be combined into a single radical.

• Rationalizing the Denominator: This process involves eliminating radicals from the denominator of a fraction by multiplying the numerator and denominator by an appropriate form of 1 (like $\frac{\sqrt{y}}{\sqrt{y}}$).

• Simplifying Fractions: Fractions are simplified by cancelling out common factors from the numerator and denominator.

• Exponent Rules: The power rule states that $a^{m} a^{n} = a^{m + n}$, and the power of a power rule states that $(a^{m})^{n} = a^{m \cdot n}$.

• Radical to Exponent Conversion: The expression $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ is used to convert between radical and exponent form.

• Product Rule for Radicals: The product rule allows us to combine radicals by multiplying the radicands when the indices are the same: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.