Simplify square root of 23/2300
This question is asking for the simplification of the radical expression, which involves the square root of a fraction. Specifically, the fraction under the square root is 23/2300, and the problem requires you to perform operations that simplify the square root of this fraction to its simplest form. This may involve factoring out perfect squares and simplifying the numbers inside and outside the square root, as well as reducing the ratio of the fraction itself.
$\sqrt{\frac{23}{2300}}$
Answer
Solution:
Simplify $\sqrt{\frac{23}{2300}}$
Step 1:
Identify and remove common factors from numerator and denominator.
Step 1.1:
Extract the factor of $23$ from the numerator.
$\sqrt{\frac{23 \times 1}{2300}}$
Step 1.2:
Eliminate identical factors in numerator and denominator.
Step 1.2.1:
Extract the factor of $23$ from the denominator.
$\sqrt{\frac{23 \times 1}{23 \times 100}}$
Step 1.2.2:
Remove the common factor of $23$.
$\sqrt{\frac{\cancel{23} \times 1}{\cancel{23} \times 100}}$
Step 1.2.3:
Simplify the fraction under the radical.
$\sqrt{\frac{1}{100}}$
Step 2:
Express $\sqrt{\frac{1}{100}}$ as a quotient of square roots.
$\frac{\sqrt{1}}{\sqrt{100}}$
Step 3:
Recognize that the square root of $1$ is $1$.
$\frac{1}{\sqrt{100}}$
Step 4:
Simplify the square root in the denominator.
Step 4.1:
Express $100$ as a square of an integer.
$\frac{1}{\sqrt{10^2}}$
Step 4.2:
Extract the square root of the perfect square.
$\frac{1}{10}$
Step 5:
Present the final result in various formats.
Exact Form: $\frac{1}{10}$
Decimal Form: $0.1$
Knowledge Notes:
To simplify a square root of a fraction, you can follow these steps:
Factorize both the numerator and the denominator to identify common factors.
Cancel out any common factors from the numerator and the denominator.
If the fraction under the square root simplifies to a fraction of perfect squares, rewrite the square root of the fraction as a fraction of square roots.
Simplify the square roots of perfect squares, as the square root of a perfect square is an integer.
If necessary, convert the simplified fraction into decimal form.
Relevant mathematical concepts include:
Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number.
Simplifying Fractions: This involves reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common factor.
Perfect Squares: Numbers that are squares of integers. Their square roots are integers.
Radical Notation: The symbol used to denote the square root or nth root of a number.
Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction.
In LaTeX, multiplication is often denoted by $\times$, square roots by $\sqrt{}$, and cancellation of common factors by $\cancel{}$.
