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

Solution:

Step 1:

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

Step 2:

Simplify the fraction inside the radical.

Step 2.1:

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

Step 2.2:

Eliminate the common factors.

Step 2.2.1:

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

Step 2.2.2:

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

Step 2.2.3:

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

Step 2.2.4:

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

Step 3:

Refine the numerator's radical.

Step 3.1:

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

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

Step 3.2:

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

Step 4:

Reduce the common $y$ terms.

Step 4.1:

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

Step 4.2:

Eliminate the common factors.

Step 4.2.1:

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

Step 4.2.2:

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

Step 4.2.3:

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

Step 4.2.4:

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

Step 4.2.5:

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

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.