Simplify ( square root of -14)/( square root of -7)
The question asks to simplify the expression where the square root of negative 14 is divided by the square root of negative 7. It involves complex numbers since the square root of a negative number is not defined in the set of real numbers. The simplification will likely involve the imaginary unit 'i', which is defined as the square root of -1. The task involves applying the rules of radicals and simplifying the expression to the simplest form possible, often involving the imaginary unit.
$\frac{\sqrt{- 14}}{\sqrt{- 7}}$
Answer
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: $1.41421356\ldots$
Knowledge Notes:
Imaginary Unit: The imaginary unit $i$ is defined as $\sqrt{-1}$. It is used to express square roots of negative numbers, where $i^2 = -1$.
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 $i$.
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.
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.
Radical Expressions: A radical expression involves roots, such as square roots. Combining and simplifying radical expressions is similar to combining and simplifying fractions.
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.
