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

Solution:

Step:1

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

Step:2

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

Step:2.1

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

Step:2.2

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

Step:2.3

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

Step:2.4

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

Step:3

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

Step:4

Multiply $\frac{\sqrt{5x}}{\sqrt{y}}$ 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: $\frac{\sqrt{5x}\sqrt{y}}{\sqrt{y}\sqrt{y}}$.

Step:5.2

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

Step:5.3

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

Step:5.4

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

Step:5.5

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

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: $\frac{\sqrt{5x}\sqrt{y}}{((y^{\frac{1}{2}}){)}^2}$.

Step:5.6.2

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

Step:5.6.3

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

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: $\frac{\sqrt{5x}\sqrt{y}}{y^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step:5.6.4.2

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

Step:5.6.5

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

Step:6

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

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: $\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$.