Simplify ( square root of 18x^3y^10)/( square root of 32xy^4)
The question is asking for the simplification of a rational expression involving square roots. Specifically, you are to simplify the square root of the product of 18, x raised to the power of 3, and y raised to the power of 10, divided by the square root of the product of 32, x, and y raised to the power of 4. This is an algebraic problem that requires knowledge of the properties of exponents and radicals to combine and reduce the expression to its simplest form.
$\frac{\sqrt{18 x^{3} y^{10}}}{\sqrt{32 x y^{4}}}$
Answer
Solution:
Step 1
Combine the square root of the numerator and the square root of the denominator into a single square root expression: $\sqrt{\frac{18x^3y^{10}}{32xy^4}}$.
Step 2
Simplify the fraction under the radical by finding and canceling out common factors.
Step 2.1
Extract the factor of 2 from the numerator: $\sqrt{\frac{2(9x^3y^{10})}{32xy^4}}$.
Step 2.2
Extract the factor of 2 from the denominator: $\sqrt{\frac{2(9x^3y^{10})}{2(16xy^4)}}$.
Step 2.3
Cancel out the common factor of 2: $\sqrt{\frac{\cancel{2}(9x^3y^{10})}{\cancel{2}(16xy^4)}}$.
Step 2.4
Simplify the expression: $\sqrt{\frac{9x^3y^{10}}{16xy^4}}$.
Step 3
Further reduce the expression by canceling out common x terms.
Step 3.1
Factor out x from the numerator: $\sqrt{\frac{x(9x^2y^{10})}{16xy^4}}$.
Step 3.2
Proceed with canceling common factors.
Step 3.2.1
Factor out x from the denominator: $\sqrt{\frac{x(9x^2y^{10})}{x(16y^4)}}$.
Step 3.2.2
Cancel the common x factor: $\sqrt{\frac{\cancel{x}(9x^2y^{10})}{\cancel{x}(16y^4)}}$.
Step 3.2.3
Simplify the expression: $\sqrt{\frac{9x^2y^{10}}{16y^4}}$.
Step 4
Cancel out common y terms.
Step 4.1
Factor out $y^4$ from the numerator: $\sqrt{\frac{y^4(9x^2y^6)}{16y^4}}$.
Step 4.2
Continue with the cancellation process.
Step 4.2.1
Factor out $y^4$ from the denominator: $\sqrt{\frac{y^4(9x^2y^6)}{y^4 \cdot 16}}$.
Step 4.2.2
Cancel the common $y^4$ factor: $\sqrt{\frac{\cancel{y^4}(9x^2y^6)}{\cancel{y^4} \cdot 16}}$.
Step 4.2.3
Simplify the expression: $\sqrt{\frac{9x^2y^6}{16}}$.
Step 5
Express $9x^2y^6$ as a square: $\left(3xy^3\right)^2$.
Step 6
Express 16 as a square: $4^2$.
Step 7
Combine the squares under the radical: $\sqrt{\left(\frac{3xy^3}{4}\right)^2}$.
Step 8
Extract the square term from under the radical, assuming all variables represent positive real numbers: $\frac{3xy^3}{4}$.
Knowledge Notes:
To simplify a radical expression involving fractions, the following knowledge points are relevant:
Radical Simplification: The process of simplifying expressions under a square root (or any root) involves finding factors that are perfect squares and simplifying them outside the radical.
Combining Radicals: When you have a fraction under a radical, you can combine the square root of the numerator and the square root of the denominator into one square root.
Factoring and Canceling: Common factors in the numerator and denominator can be factored out and canceled. This is based on the property that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
Simplifying Fractions: Before simplifying the radical, it's often helpful to simplify the fraction itself by canceling common factors.
Square of a Binomial: Recognizing that a term like $(3xy^3)^2$ is a perfect square allows us to simplify the square root of that term to just the term itself.
Assumption of Positive Real Numbers: When pulling terms out from under a radical, it is typically assumed that the variables represent positive real numbers to avoid dealing with absolute values.
By applying these principles, we can simplify the given radical expression step by step.
