Simplify ( square root of xyz)/( square root of yyz)

Solution:

Step:1

Merge $\sqrt{xyz}$ with $\sqrt{yyz}$ under a single radical to get $\sqrt{\frac{xyz}{yyz}}$.

Step:2

Simplify the fraction $\frac{xyz}{yyz}$ by removing common factors.

Step:2.1

Eliminate the shared factor $y$ to obtain $\sqrt{\frac{x \cancel{y} z}{\cancel{y} y z}}$.

Step:2.2

Reformulate the expression as $\sqrt{\frac{xz}{yz}}$.

Step:3

Remove the common $z$ factor.

Step:3.1

Strike out the shared $z$ factor to get $\sqrt{\frac{x \cancel{z}}{y \cancel{z}}}$.

Step:3.2

Rephrase the expression as $\sqrt{\frac{x}{y}}$.

Step:4

Convert $\sqrt{\frac{x}{y}}$ to the form $\frac{\sqrt{x}}{\sqrt{y}}$.

Step:5

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

Step:6

Simplify the expression in the denominator.

Step:6.1

Multiply the numerators and denominators by $\sqrt{y}$ to get $\frac{\sqrt{x} \sqrt{y}}{\sqrt{y} \sqrt{y}}$.

Step:6.2

Express $\sqrt{y}$ as $\left(\sqrt{y}\right)^{1}$.

Step:6.3

Repeat the expression of $\sqrt{y}$ as $\left(\sqrt{y}\right)^{1}$.

Step:6.4

Apply the exponent rule $a^{m} a^{n} = a^{m + n}$ to combine the powers of $\sqrt{y}$.

Step:6.5

Sum the exponents to get $\frac{\sqrt{x} \sqrt{y}}{\left(\sqrt{y}\right)^{2}}$.

Step:6.6

Transform $\left(\sqrt{y}\right)^{2}$ back to $y$.

Step:6.6.1

Rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$.

Step:6.6.2

Apply the power rule $\left(a^{m}\right)^{n} = a^{m n}$ to get $\frac{\sqrt{x} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$.

Step:6.6.3

Combine the fraction $\frac{1}{2}$ with $2$ to simplify.

Step:6.6.4

Reduce the fraction by cancelling out the common factor of $2$.

Step:6.6.4.1

Strike through the common $2$ to get $\frac{\sqrt{x} \sqrt{y}}{y^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step:6.6.4.2

Reformulate the expression as $\frac{\sqrt{x} \sqrt{y}}{y^{1}}$.

Step:6.6.5

Simplify the expression to $\frac{\sqrt{x} \sqrt{y}}{y}$.

Step:7

Combine the radicals using the product rule to get $\frac{\sqrt{xy}}{y}$.

Knowledge Notes:

To simplify the expression $\frac{\sqrt{xyz}}{\sqrt{yyz}}$, we follow these steps:

1. Combining Radicals: Radicals can be combined under a single radical sign when they have the same index (the root number). In this case, both are square roots.

2. Simplifying Fractions: When simplifying fractions, we look for common factors in the numerator and denominator that can be cancelled out.

3. Rationalizing the Denominator: When a radical is present in the denominator, we multiply the fraction by a form of 1 that will eliminate the radical in the denominator. This is often done by multiplying by the conjugate or by a radical that will square out the denominator.

4. Properties of Radicals and Exponents: We use properties such as $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ and $a^{m} a^{n} = a^{m + n}$ to simplify expressions involving radicals and exponents.

5. Product Rule for Radicals: The product rule for radicals states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, provided that a and b are nonnegative.

By following these principles, we can simplify radical expressions and rationalize denominators to obtain a simplified form.