Simplify ( square root of 28y^5)/( square root of 7y)

Solution:

Step 1:

Merge $\sqrt{28 y^{5}}$ and $\sqrt{7y}$ into a single square root expression. $\frac{\sqrt{28 y^{5} / 7y}}{1}$

Step 2:

Simplify the fraction inside the radical.

Step 2.1:

Extract the factor of 7 from $28 y^{5}$. $\frac{\sqrt{7 \cdot 4 y^{5} / 7}}{1}$

Step 2.2:

Eliminate the common factors.

Step 2.2.1:

Separate the factor of 7 from the denominator. $\frac{\sqrt{7 \cdot 4 y^{5} / (7 \cdot 1)}}{1}$

Step 2.2.2:

Remove the common factor of 7. $\frac{\sqrt{\cancel{7} \cdot 4 y^{5} / \cancel{7} \cdot 1}}{1}$

Step 2.2.3:

Reformulate the expression. $\frac{\sqrt{4 y^{5}}}{1}$

Step 2.2.4:

Divide $4 y^{5}$ by 1. $\sqrt{4 y^{5}}$

Step 3:

Refine the numerator's radical.

Step 3.1:

Express $4 y^{5}$ as $(2 y^{2})^{2} \cdot y$.

Step 3.1.1:

Represent 4 as $2^{2}$. $\frac{\sqrt{2^{2} \cdot y^{5}}}{1}$

Step 3.1.2:

Factor $y^{4}$ from the expression. $\frac{\sqrt{2^{2} \cdot (y^{4} \cdot y)}}{1}$

Step 3.1.3:

Rewrite $y^{4}$ as $(y^{2})^{2}$. $\frac{\sqrt{2^{2} \cdot ((y^{2})^{2} \cdot y)}}{1}$

Step 3.1.4:

Rephrase $2^{2} \cdot (y^{2})^{2}$ as $(2 y^{2})^{2}$. $\sqrt{(2 y^{2})^{2} \cdot y}$

Step 3.2:

Extract terms from under the square root. $2 y^{2} \sqrt{y}$

Step 4:

Reduce the common $y$ terms.

Step 4.1:

Factor out $y$ from $2 y^{2} \sqrt{y}$. $\frac{y \cdot (2 y \sqrt{y})}{y}$

Step 4.2:

Eliminate the common factors.

Step 4.2.1:

Raise $y$ to the first power. $\frac{y \cdot (2 y \sqrt{y})}{y^{1}}$

Step 4.2.2:

Separate $y$ from $y^{1}$. $\frac{y \cdot (2 y \sqrt{y})}{y \cdot 1}$

Step 4.2.3:

Remove the common $y$ factor. $\frac{\cancel{y} \cdot (2 y \sqrt{y})}{\cancel{y} \cdot 1}$

Step 4.2.4:

Reformulate the expression. $\frac{2 y \sqrt{y}}{1}$

Step 4.2.5:

Divide $2 y \sqrt{y}$ by 1. $2 y \sqrt{y}$

Knowledge Notes:

The problem involves simplifying a radical expression, which requires understanding of square roots, factoring, and simplifying fractions. Here are the relevant knowledge points:

1. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, $\sqrt{9} = 3$ because $3 \times 3 = 9$.

2. Radicals: A radical expression is an expression that includes a root, such as a square root, cube root, etc. Simplifying radicals involves factoring numbers or expressions and pulling out pairs of numbers or variables.

3. Factoring: Factoring is the process of breaking down a number or expression into its component factors. For example, $28 = 7 \times 4$.

4. Simplifying Fractions: To simplify a fraction, divide the numerator and the denominator by their greatest common factor.

5. Algebraic Manipulation: This involves rewriting expressions in different but equivalent forms to simplify them or make them more understandable. This can include expanding, factoring, and canceling common factors.

6. Properties of Exponents: When simplifying expressions involving exponents, remember that $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$.

7. Combining Radicals: When dealing with square roots, you can combine them under one radical if they are being multiplied or divided. For example, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

By applying these principles, one can simplify complex radical expressions step by step.