Simplify ( square root of 100)/(2 square root of 25)
The question asks to simplify a given mathematical expression that involves the operation of division between two square root terms. The numerator of the expression is the square root of 100, and the denominator is two times the square root of 25. You are expected to perform the appropriate arithmetic operations and simplify the expression to the simplest form possible, which likely involves calculating the square roots and reducing the expression to a rational number or a simpler radical form.
$\frac{\sqrt{100}}{2 \sqrt{25}}$
Answer
Solution:
Step 1:
Merge the square roots in the numerator and denominator to form a single square root: $\frac{\sqrt{\frac{100}{25}}}{2}$.
Step 2:
Identify and remove the common factors between the numerator and the denominator.
Step 2.1:
Express 100 as a multiple of 25: $\frac{\sqrt{\frac{25 \cdot 4}{25}}}{2}$.
Step 2.2:
Eliminate the common factors.
Step 2.2.1:
Factor out 25 from the denominator: $\frac{\sqrt{\frac{25 \cdot 4}{25 \cdot 1}}}{2}$.
Step 2.2.2:
Cross out the 25 in both the numerator and the denominator: $\frac{\sqrt{\frac{\cancel{25} \cdot 4}{\cancel{25} \cdot 1}}}{2}$.
Step 2.2.3:
Reformulate the expression: $\frac{\sqrt{\frac{4}{1}}}{2}$.
Step 2.2.4:
Divide 4 by 1: $\frac{\sqrt{4}}{2}$.
Step 3:
Simplify the square root in the numerator.
Step 3.1:
Represent 4 as $2^2$: $\frac{\sqrt{2^2}}{2}$.
Step 3.2:
Extract the square root of the perfect square: $\frac{2}{2}$.
Step 4:
Perform the division of 2 by 2 to get the final answer: $1$.
Knowledge Notes:
To simplify the expression $\frac{\sqrt{100}}{2\sqrt{25}}$, we use several mathematical concepts:
Radical Simplification: The square root of a fraction can be simplified by taking the square root of the numerator and the denominator separately.
Common Factors: If the numerator and denominator of a fraction share a common factor, it can be canceled out to simplify the fraction.
Perfect Squares: A perfect square is a number that can be expressed as the square of an integer. The square root of a perfect square is always an integer.
Simplifying Square Roots: The square root of a perfect square is simplified by finding the number that, when squared, equals the number under the radical.
Division: When a number is divided by itself, the result is 1.
In this problem, we first combined the square roots into a single radical and then canceled out the common factor of 25. We recognized that 4 is a perfect square and simplified the square root of 4 to 2. Finally, we divided 2 by 2 to get the answer of 1.
