Simplify square root of (450x^7y^5)/(2x^5y^3)
The given question asks to simplify the square root of a fraction, specifically the square root of (450x^7y^5)/(2x^5y^3). This involves performing the operation of taking the square root on both the numerator and the denominator of the fraction, as well as simplifying the resulting expression by cancelling out common factors and applying the laws of exponents for the variables x and y. The simplification process should aim to reduce the fraction to its simplest form within the radical.
$\sqrt{\frac{450 x^{7} y^{5}}{2 x^{5} y^{3}}}$
Answer
Solution:
Simplification Process:
Step 1: Simplify the fraction $\frac{450 x^{7} y^{5}}{2 x^{5} y^{3}}$ by removing common factors.
Step 1.1: Extract the factor of $2$ from the numerator: $\sqrt{\frac{2 \cdot 225 x^{7} y^{5}}{2 x^{5} y^{3}}}$ Step 1.2: Extract the factor of $2$ from the denominator: $\sqrt{\frac{2 \cdot 225 x^{7} y^{5}}{2 \cdot x^{5} y^{3}}}$ Step 1.3: Eliminate the common factor of $2$: $\sqrt{\frac{\cancel{2} \cdot 225 x^{7} y^{5}}{\cancel{2} \cdot x^{5} y^{3}}}$ Step 1.4: Present the simplified expression: $\sqrt{\frac{225 x^{7} y^{5}}{x^{5} y^{3}}}$
Step 2: Reduce the powers of $x$ by cancelling out common terms.
Step 2.1: Factor out $x^{5}$ from the numerator: $\sqrt{\frac{x^{5} \cdot 225 x^{2} y^{5}}{x^{5} y^{3}}}$ Step 2.2: Remove common factors of $x^{5}$.
Step 2.2.1: Factor out $x^{5}$ from the denominator: $\sqrt{\frac{x^{5} \cdot 225 x^{2} y^{5}}{x^{5} \cdot y^{3}}}$ Step 2.2.2: Cancel the common $x^{5}$ term: $\sqrt{\frac{\cancel{x^{5}} \cdot 225 x^{2} y^{5}}{\cancel{x^{5}} \cdot y^{3}}}$ Step 2.2.3: Rewrite the expression: $\sqrt{\frac{225 x^{2} y^{5}}{y^{3}}}$
Step 3: Simplify the powers of $y$ by cancelling out common terms.
Step 3.1: Factor out $y^{3}$ from the numerator: $\sqrt{\frac{y^{3} \cdot 225 x^{2} y^{2}}{y^{3}}}$ Step 3.2: Remove common factors of $y^{3}$.
Step 3.2.1: Multiply the denominator by $1$: $\sqrt{\frac{y^{3} \cdot 225 x^{2} y^{2}}{y^{3} \cdot 1}}$ Step 3.2.2: Cancel the common $y^{3}$ term: $\sqrt{\frac{\cancel{y^{3}} \cdot 225 x^{2} y^{2}}{\cancel{y^{3}} \cdot 1}}$ Step 3.2.3: Rewrite the expression: $\sqrt{\frac{225 x^{2} y^{2}}{1}}$ Step 3.2.4: Divide $225 x^{2} y^{2}$ by $1$: $\sqrt{225 x^{2} y^{2}}$
Step 4: Express $225 x^{2} y^{2}$ as a perfect square: $\sqrt{(15 x y)^{2}}$
Step 5: Extract the square root of the perfect square: $15 x y$
Knowledge Notes:
To simplify a square root of a fraction, we follow these steps:
Factorization: Break down both the numerator and denominator into their prime factors or common factors to simplify the fraction.
Cancellation: Cancel out any common factors in the numerator and denominator. This includes common numerical factors as well as any common variables raised to a power.
Simplifying Powers: When variables are raised to powers, you can simplify by subtracting the powers if they are in the form of a fraction (using the rule $x^{a}/x^{b} = x^{a-b}$).
Square Roots: When simplifying square roots, if the argument of the square root is a perfect square (like $a^{2}$), then the square root is simply the base of the power (like $a$).
Rationalizing the Denominator: If necessary, we make sure that there are no radicals in the denominator. However, in this problem, after simplification, the denominator is $1$, so this step is not needed.
Assumption of Positive Real Numbers: When simplifying square roots, we generally assume that the variables represent positive real numbers to avoid dealing with complex numbers or absolute values.
