Simplify square root of 100/64
The question asks for the simplification of the square root of the fraction 100/64. This involves calculating the square root of both the numerator (100) and the denominator (64) separately to simplify the fraction under the square root, to ultimately find the simplified form of the original square root expression.
$\sqrt{\frac{100}{64}}$
Answer
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:
Common Factor Reduction: If the numerator and the denominator of a fraction have common factors, they can be cancelled out to simplify the fraction.
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., $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
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 $4^2 = 16$.
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 $x$.
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.
