Simplify square root of (16y^7)/(x^8)
The question asks for the simplification of a radical expression, specifically the square root of a quotient involving exponents. It involves taking the square root of the entire fraction where the numerator is 16 times y to the 7th power, and the denominator is x to the 8th power. The simplification process typically involves finding perfect squares within the radical and applying the property that the square root of a quotient is equal to the quotient of the square roots. It may also require the use of exponent rules to simplify the variable expressions within the radical.
$\sqrt{\frac{16 y^{7}}{x^{8}}}$
Answer
Solution:
Step 1
Express $\frac{16y^7}{x^8}$ as $\left(\frac{4y^3}{x^4}\right)^2 \cdot y$.
Step 1.1
Extract the square of the perfect square $4y^3$ from $16y^7$ to get $\sqrt{\frac{(4y^3)^2 \cdot y}{x^8}}$.
Step 1.2
Extract the square of the perfect square $x^4$ from $x^8$ to form $\sqrt{\frac{(4y^3)^2 \cdot y}{(x^4)^2 \cdot 1}}$.
Step 1.3
Reorganize the fraction $\frac{(4y^3)^2 \cdot y}{(x^4)^2 \cdot 1}$ to $\sqrt{\left(\frac{4y^3}{x^4}\right)^2 \cdot y}$.
Step 2
Remove terms from under the square root to obtain $\frac{4y^3}{x^4} \sqrt{y}$.
Step 3
Merge $\frac{4y^3}{x^4}$ with $\sqrt{y}$ to finalize the simplification as $\frac{4y^3\sqrt{y}}{x^4}$.
Knowledge Notes:
To simplify the square root of a fraction, we can use the property that the square root of a product is equal to the product of the square roots of the individual factors. We start by identifying perfect squares within the numerator and denominator that can be factored out. In this case, $16y^7$ can be rewritten as $(4y^3)^2 \cdot y$ because $16 = 4^2$ and $y^7 = (y^3)^2 \cdot y$. Similarly, $x^8$ can be rewritten as $(x^4)^2$ because $x^8 = (x^4)^2$.
Once we have factored out the perfect squares, we can take the square root of the entire expression. The square root of a perfect square is simply the base of the square, so $(4y^3)^2$ under the square root becomes $4y^3$, and $(x^4)^2$ under the square root becomes $x^4$. The remaining term under the square root, which is not a perfect square, stays within the square root.
Finally, we combine the terms that are outside the square root with those that remain inside to complete the simplification. It is important to ensure that the expression is simplified as much as possible, which means that there should be no perfect squares left under the square root and no factors common to the numerator and denominator that can be canceled out.
