Simplify ( square root of 99xy^3)/( square root of 9x)
The problem is asking for the simplification of a radical fraction. Specifically, it involves taking the square root of 99 times x times y to the power of 3, and dividing that by the square root of 9 times x. The goal is to simplify the expression by using properties of square roots and the laws of exponents to factor and reduce the fraction to its simplest form, while also simplifying the radicals involved.
$\frac{\sqrt{99 x y^{3}}}{\sqrt{9 x}}$
Answer
Solution:
Step 1:
Merge the square roots $\sqrt{99xy^3}$ and $\sqrt{9x}$ into a single square root expression: $\sqrt{\frac{99xy^3}{9x}}$.
Step 2:
Simplify the fraction $\frac{99xy^3}{9x}$ by eliminating common factors.
Step 2.1:
Extract the factor of $9$ from $99xy^3$: $\sqrt{\frac{9(11xy^3)}{9x}}$.
Step 2.2:
Extract the factor of $9$ from $9x$: $\sqrt{\frac{9(11xy^3)}{9(x)}}$.
Step 2.3:
Remove the common factor of $9$: $\sqrt{\frac{\cancel{9}(11xy^3)}{\cancel{9}x}}$.
Step 2.4:
Reformulate the simplified expression: $\sqrt{\frac{11xy^3}{x}}$.
Step 3:
Eliminate the common $x$ factor from the numerator and denominator.
Step 3.1:
Remove the common $x$: $\sqrt{\frac{11\cancel{x}y^3}{\cancel{x}}}$.
Step 3.2:
Compute $11y^3$ divided by $1$: $\sqrt{11y^3}$.
Step 4:
Express $11y^3$ as $y^2 \cdot (11y)$.
Step 4.1:
Isolate the $y^2$ term: $\sqrt{11(y^2y)}$.
Step 4.2:
Rearrange $11$ and $y^2$: $\sqrt{y^2 \cdot 11y}$.
Step 4.3:
Introduce parentheses: $\sqrt{y^2 \cdot (11y)}$.
Step 5:
Extract terms from under the square root: $y\sqrt{11y}$.
Knowledge Notes:
To simplify a radical expression involving square roots, follow these steps:
Combining Radicals: When you have a fraction under a square root, you can combine them into a single square root.
Simplifying Fractions: Simplify the fraction inside the square root by canceling out common factors.
Extracting Square Factors: If there's a perfect square under the radical, you can take the square root of that term and move it outside the radical.
Rationalizing the Denominator: If the denominator of a fraction under a radical is not a perfect square, you might need to multiply the numerator and the denominator by a term that will make the denominator a perfect square.
Simplifying Square Roots: After simplifying, if there are still square roots, look for any terms that can be taken out of the square root (terms that are perfect squares).
Algebraic Manipulation: Use algebraic rules to factor, expand, or combine like terms as needed to simplify the expression further.
In this particular problem, we simplified the square root of a fraction by canceling out common factors both in the numerator and the denominator and then extracting the square root of a perfect square (in this case, $y^2$) from under the radical.
