Simplify square root of (101x^14)/(25y^8)
The given problem is asking for the simplification of a radical expression. The expression contains a square root of a fraction, where the numerator is 101 times the variable x raised to the 14th power and the denominator is 25 times the variable y raised to the 8th power. The task is to simplify this square root by factoring perfect squares from inside the radical and reducing the expression to simplest form, if possible.
$\sqrt{\frac{101 x^{14}}{25 y^{8}}}$
Answer
Solution:
Step 1
Reexpress $\frac{101 x^{14}}{25 y^{8}}$ as $\left(\frac{x^{7}}{5 y^{4}}\right)^{2} \cdot 101$.
Step 1.1
Extract the perfect square $x^{14}$ as $(x^{7})^{2}$ from $101 x^{14}$ to get $\sqrt{\frac{(x^{7})^{2} \cdot 101}{25 y^{8}}}$.
Step 1.2
Extract the perfect square $25 y^{8}$ as $(5 y^{4})^{2}$ to get $\sqrt{\frac{(x^{7})^{2} \cdot 101}{(5 y^{4})^{2} \cdot 1}}$.
Step 1.3
Rearrange the fraction to $\sqrt{\left(\frac{x^{7}}{5 y^{4}}\right)^{2} \cdot 101}$.
Step 2
Remove terms from under the square root, yielding $\frac{x^{7}}{5 y^{4}} \cdot \sqrt{101}$.
Step 3
Merge the terms $\frac{x^{7}}{5 y^{4}}$ and $\sqrt{101}$ to form the simplified expression $\frac{x^{7} \cdot \sqrt{101}}{5 y^{4}}$.
Knowledge Notes:
The problem involves simplifying a square root of a fraction that contains variables raised to powers. The key steps in the simplification process include:
Recognizing perfect squares: In algebra, a perfect square is a number or an expression that is the square of an integer or a more complex expression. In this problem, $x^{14}$ and $25 y^{8}$ are perfect squares because they can be written as $(x^{7})^{2}$ and $(5 y^{4})^{2}$ respectively.
Simplifying square roots: The square root of a perfect square is simply the number or expression that was squared. For example, $\sqrt{(x^{7})^{2}} = x^{7}$.
Rationalizing the denominator: This involves rewriting the square root of a fraction in a form that has no square root in the denominator. In this problem, the denominator is already a perfect square, so no additional rationalization is needed.
Combining terms: After simplifying the square root, the terms outside the square root are combined to form the final simplified expression.
Properties of radicals: When simplifying radicals, it's important to remember that $\sqrt{a^2} = a$ if $a \geq 0$, and that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
Properties of exponents: The problem also uses the property that $(a^m)^n = a^{mn}$, which is used to simplify expressions with powers raised to additional powers.
In this problem, the expression is simplified by factoring out perfect squares and then taking the square root of those perfect squares, while leaving the non-perfect square (101 in this case) under the radical. The final result is a simplified expression that is easier to work with and understand.
