Simplify ( square root of 18xy^3)/( square root of 9x)
The given problem is asking to simplify a fraction where both the numerator and the denominator contain square roots with algebraic expressions. Specifically, the numerator is the square root of the product of 18, x, and y cubed (y^3), while the denominator is the square root of the product of 9 and x. The objective is to perform algebraic simplification to possibly reduce the expression to its simplest form by canceling common factors and applying the properties of square roots and algebraic manipulation.
$\frac{\sqrt{18 x y^{3}}}{\sqrt{9 x}}$
Answer
Solution:
Step 1:
Merge the square roots into a single expression: $\sqrt{\frac{18xy^3}{9x}}$.
Step 2:
Simplify the fraction under the radical by removing common factors.
Step 2.1:
Extract the factor of 9 from the numerator: $\sqrt{\frac{9(2xy^3)}{9x}}$.
Step 2.2:
Extract the factor of 9 from the denominator: $\sqrt{\frac{9(2xy^3)}{9(x)}}$.
Step 2.3:
Eliminate the common factor of 9: $\sqrt{\frac{\cancel{9}(2xy^3)}{\cancel{9}x}}$.
Step 2.4:
Present the simplified expression: $\sqrt{\frac{2xy^3}{x}}$.
Step 3:
Further reduce the expression by canceling out the common x factor.
Step 3.1:
Remove the common x factor: $\sqrt{\frac{2\cancel{x}y^3}{\cancel{x}}}$.
Step 3.2:
Simplify the expression to: $\sqrt{2y^3}$.
Step 4:
Reorganize $2y^3$ to highlight the perfect square factor.
Step 4.1:
Separate $y^2$ from the expression: $\sqrt{2(y^2y)}$.
Step 4.2:
Rearrange the terms: $\sqrt{y^2 \cdot 2y}$.
Step 4.3:
Group the terms inside the radical: $\sqrt{y^2 \cdot (2y)}$.
Step 5:
Extract the perfect square term from under the radical to obtain the final simplified form: $y\sqrt{2y}$.
Knowledge Notes:
To simplify a radical expression involving square roots, follow these steps:
Combine the radicals if possible by placing them under a common radical sign and simplifying the resulting expression.
Factor out any perfect squares from under the radical to simplify the expression further.
Cancel out any common factors in the numerator and denominator, if the radical is a fraction.
If there are variables under the radical, separate them into terms that are perfect squares and those that are not to simplify the expression.
Extract any perfect squares from under the radical to simplify the radical expression.
In the given problem, we used these steps to simplify the expression $\sqrt{\frac{18xy^3}{9x}}$. We combined the radicals, canceled common factors, and extracted the perfect square $y^2$ from under the radical to arrive at the simplified expression $y\sqrt{2y}$.
