Simplify ( square root of -14)/( square root of -7)

Solution:

Step 1: Extract the imaginary unit $i$ from the square roots.

Step 1.1

Express $-14$ as $-1\times 14$. Thus, we have $\frac{\sqrt{-1\times 14}}{\sqrt{-7}}$.

Step 1.2

Separate the square root of the product into the product of square roots: $\frac{\sqrt{-1}\cdot \sqrt{14}}{\sqrt{-7}}$.

Step 1.3

Replace $\sqrt{-1}$ with $i$: $\frac{i\cdot \sqrt{14}}{\sqrt{-7}}$.

Step 1.4

Express $-7$ as $-1\times 7$: $\frac{i\cdot \sqrt{14}}{\sqrt{-1\times 7}}$.

Step 1.5

Again, separate the square root of the product into the product of square roots: $\frac{i\cdot \sqrt{14}}{\sqrt{-1}\cdot \sqrt{7}}$.

Step 1.6

Replace $\sqrt{-1}$ with $i$ in the denominator: $\frac{i\cdot \sqrt{14}}{i\cdot \sqrt{7}}$.

Step 2: Eliminate the common imaginary unit $i$.

Step 2.1

Remove the common $i$ from numerator and denominator: $\frac{\cancel{i}\cdot \sqrt{14}}{\cancel{i}\cdot \sqrt{7}}$.

Step 2.2

Simplify the expression to $\frac{\sqrt{14}}{\sqrt{7}}$.

Step 3: Combine the square roots.

Combine the square roots into one: $\sqrt{\frac{14}{7}}$.

Step 4: Simplify the radical.

Divide the numbers inside the square root: $\sqrt{2}$.

Step 5: Present the final result in various forms.

Exact Form: $\sqrt{2}$ Decimal Form: $1.41421356\dots$

Knowledge Notes:

1. Imaginary Unit: The imaginary unit $i$ is defined as $\sqrt{-1}$. It is used to express square roots of negative numbers, where $i^2=-1$.

2. Square Roots: The square root of a number $x$ is a number $y$ such that $y^2=x$. For negative numbers, the square root involves the imaginary unit $i$.

3. Simplifying Square Roots: To simplify the square root of a negative number, the imaginary unit $i$ is extracted, and the remaining positive part is treated as a standard square root.

4. Rationalizing Complex Fractions: When a fraction involves complex numbers or imaginary units, it can often be simplified by eliminating common factors in the numerator and denominator.

5. Radical Expressions: A radical expression involves roots, such as square roots. Combining and simplifying radical expressions is similar to combining and simplifying fractions.

6. Exact vs. Decimal Form: The exact form of a number is its expression in terms of radicals or simple fractions, while the decimal form is an approximate value that represents the number using decimal digits.