Simplify ( square root of 49y^7)/(x^10)

Solution:

Step 1:

Express $49y^7$ as $(7y^3{)}^{2}y$.

Step 1.1:

Represent $49$ as $7^2$. Thus, we have $\frac{\sqrt{7^2{y}^7}}{x^{10}}$.

Step 1.2:

Extract $y^6$ from the expression. We get $\frac{\sqrt{7^2(y^{6}y)}}{x^{10}}$.

Step 1.3:

Express $y^6$ as $(y^3{)}^2$. This gives us $\frac{\sqrt{7^2((y^3{)}^{2}y)}}{x^{10}}$.

Step 1.4:

Combine $7^2$ and $(y^3{)}^2$ to form $(7y^3{)}^2$. The expression now is $\frac{\sqrt{(7y^3{)}^{2}y}}{x^{10}}$.

Step 2:

Extract terms from under the square root. The simplified form is $\frac{7y^3\sqrt{y}}{x^{10}}$.

Knowledge Notes:

To simplify the given expression $\frac{\sqrt{49y^7}}{x^{10}}$, we need to apply properties of radicals and exponents. Here are the relevant knowledge points:

1. Square Root of a Square: The square root of a square number or a variable raised to an even power is the base of that square. For example, $\sqrt{a^2}=a$.

2. Exponent Rules: When multiplying powers with the same base, you add the exponents, and when taking a power of a power, you multiply the exponents.

3. Simplifying Radicals: To simplify a radical expression, you can factor out squares, cubes, etc., from under the radical sign.

4. Rationalizing the Denominator: In this problem, we don't need to rationalize the denominator, but it's a common step in simplifying radical expressions.

5. Combining Like Terms: Terms with the same variables and exponents can be combined using addition or subtraction.

In the solution, we first rewrite $49$ as $7^2$ and recognize that $y^7$ can be expressed as $y^{6}y$ or $(y^3{)}^{2}y$. This allows us to take $7y^3$ out from under the square root, simplifying the expression to $\frac{7y^3\sqrt{y}}{x^{10}}$.