Simplify square root of 100/64

Solution:

Simplification Process:

Step 1:
Identify and remove any common factors from the numerator and denominator.
Step 1.1:
Extract the factor of $4$ from the numerator. $\sqrt{\frac{4\cdot 25}{64}}$
Step 1.2:
Eliminate identical factors in both the numerator and the denominator.
Step 1.2.1:
Extract the factor of $4$ from the denominator. $\sqrt{\frac{4\cdot 25}{4\cdot 16}}$
Step 1.2.2:
Remove the common factor of $4$. $\sqrt{\frac{\cancel{4}\cdot 25}{\cancel{4}\cdot 16}}$
Step 1.2.3:
Simplify the fraction inside the radical. $\sqrt{\frac{25}{16}}$

Step 2:
Separate the square root of the fraction into the square root of the numerator and the square root of the denominator. $\frac{\sqrt{25}}{\sqrt{16}}$

Step 3:
Simplify the square root of the numerator.
Step 3.1:
Express $25$ as a square of an integer. $\frac{\sqrt{5^2}}{\sqrt{16}}$
Step 3.2:
Extract the square root of the perfect square. $\frac{5}{\sqrt{16}}$

Step 4:
Simplify the square root of the denominator.
Step 4.1:
Express $16$ as a square of an integer. $\frac{5}{\sqrt{4^2}}$
Step 4.2:
Extract the square root of the perfect square. $\frac{5}{4}$

Step 5:
Present the final result in various formats.
Exact Form: $\frac{5}{4}$
Decimal Form: $1.25$
Mixed Number Form: $1\frac{1}{4}$

Knowledge Notes:

To simplify a square root of a fraction, you can apply the following knowledge points:

1. Common Factor Reduction: If the numerator and the denominator of a fraction have common factors, they can be cancelled out to simplify the fraction.

2. Square Root of a Fraction: The square root of a fraction can be separated into the square root of the numerator divided by the square root of the denominator, i.e., $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$.

3. Square Roots of Perfect Squares: The square root of a perfect square is the integer whose square is the number under the radical. For example, $\sqrt{25}=5$ and $\sqrt{16}=4$ because $5^2=25$ and $4^2=16$.

4. Simplification of Radicals: When simplifying radicals, any factor inside the radical that is a perfect square can be taken out of the radical. This is based on the property that $\sqrt{x^2}=x$ for any positive real number $x$.

5. Fractional Forms: A fraction can be expressed in different forms, such as an exact fraction, a decimal, or a mixed number. The choice of form depends on the context and the preference for representation.