Simplify square root of (16x^3y^7)/(4z^6)
The question is asking to perform the operation of simplifying the square root of a given rational expression. The expression consists of a fraction under the square root sign, with the numerator being 16x^3y^7 and the denominator being 4z^6. To simplify it, one would need to perform simplification of both the fraction and the square root, which involves factoring out squares from the radicand to simplify the square root and reducing the fraction to its simplest form if possible. The result should be an expression in a simpler form, potentially with some variables inside the square root and some outside, depending on whether those variables' exponents are affected by the square root simplification.
$\sqrt{\frac{16 x^{3} y^{7}}{4 z^{6}}}$
Answer
Solution:
Step:1
Simplify the fraction $\frac{16 x^{3} y^{7}}{4 z^{6}}$ by removing common factors.
Step:1.1
Extract $4$ from the numerator $16 x^{3} y^{7}$ to get $\sqrt{\frac{4 \cdot (4 x^{3} y^{7})}{4 z^{6}}}$.
Step:1.2
Extract $4$ from the denominator $4 z^{6}$ to get $\sqrt{\frac{4 \cdot (4 x^{3} y^{7})}{4 \cdot (z^{6})}}$.
Step:1.3
Eliminate the common factor of $4$ to obtain $\sqrt{\frac{\cancel{4} \cdot (4 x^{3} y^{7})}{\cancel{4} z^{6}}}$.
Step:1.4
Reformulate the simplified expression as $\sqrt{\frac{4 x^{3} y^{7}}{z^{6}}}$.
Step:2
Express $\frac{4 x^{3} y^{7}}{z^{6}}$ as $\left(\frac{2 x y^{3}}{z^{3}}\right)^{2} \cdot xy$.
Step:2.1
Identify the perfect square $\left(2 x y^{3}\right)^{2}$ within $4 x^{3} y^{7}$ to get $\sqrt{\frac{\left(2 x y^{3}\right)^{2} \cdot xy}{z^{6}}}$.
Step:2.2
Identify the perfect square $\left(z^{3}\right)^{2}$ within $z^{6}$ to get $\sqrt{\frac{\left(2 x y^{3}\right)^{2} \cdot xy}{\left(z^{3}\right)^{2} \cdot 1}}$.
Step:2.3
Rearrange the fraction to $\sqrt{\left(\frac{2 x y^{3}}{z^{3}}\right)^{2} \cdot xy}$.
Step:3
Extract terms from under the square root to get $\frac{2 x y^{3}}{z^{3}} \cdot \sqrt{xy}$.
Step:4
Combine the extracted terms with the square root to finalize the expression as $\frac{2 x y^{3} \cdot \sqrt{xy}}{z^{3}}$.
Knowledge Notes:
To simplify a square root of a fraction, follow these steps:
Factorization: Break down the numerator and the denominator into their prime factors or identify common factors that can be simplified.
Simplification: Cancel out common factors from the numerator and the denominator.
Extraction of Perfect Squares: Identify and extract perfect square factors from under the square root to simplify the radical expression.
Rearrangement: Rearrange the expression to separate the terms that are under the radical from those that are not.
Combination: Combine the simplified terms to form the final expression.
In the context of this problem, the steps involve factoring out common terms, recognizing and extracting perfect squares, and simplifying the square root expression. The use of algebraic properties, such as the distributive property and the property of radicals that allows the extraction of perfect squares, is essential in this process.
