Problem 58
Question
Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \frac{\sqrt{-72}}{\sqrt{-6}} $$
Step-by-Step Solution
Verified Answer
The expression simplifies to \(2\sqrt{3}\).
1Step 1: Express in terms of imaginary unit i
The square root of a negative number can be written as the product of the imaginary unit \(i\) and the square root of the opposite positive number. Thus, \(\sqrt{-72}\) becomes \(\sqrt{72}i\) and \(\sqrt{-6}\) becomes \(\sqrt{6}i\).
2Step 2: Substitute the expressions in terms of i
Substitute the expressions from Step 1 into the original fraction: \[\frac{\sqrt{72}i}{\sqrt{6}i}.\]
3Step 3: Simplify the fraction
First, cancel out the \(i\) from the numerator and the denominator:\[\frac{\sqrt{72}i}{\sqrt{6}i} = \frac{\sqrt{72}}{\sqrt{6}}.\]Next, simplify \(\frac{\sqrt{72}}{\sqrt{6}}\) by expressing it as \(\sqrt{\frac{72}{6}} = \sqrt{12}\).
4Step 4: Simplify \(\sqrt{12}\)
Simplify \(\sqrt{12}\) by factorizing 12 into its prime factors: \(12 = 4 \times 3\). Then, \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}.\)[
Key Concepts
Imaginary UnitSquare RootsSimplification Steps
Imaginary Unit
The imaginary unit, denoted by the letter \(i\), is a fundamental concept when working with complex numbers. It is defined as the square root of -1, which is a number that cannot exist within the set of real numbers since real numbers do not have square roots of negative values.
Hence, the introduction of \(i\) allows us to express the square roots of negative numbers. When you see \(\sqrt{-x}\) for some positive number \(x\), you can rewrite this as \(\sqrt{x}i\). This is because \(i^2 = -1\), making \(i\) a bridge from real to imaginary number calculations.
The imaginary unit is essential for transforming and simplifying operations involving square roots of negative numbers. In problems such as \(\sqrt{-72}\) or \(\sqrt{-6}\), replace the negative sign with \(i\), giving \(\sqrt{72}i\) and \(\sqrt{6}i\). This is the key in expressing calculations that result in imaginary numbers.
Hence, the introduction of \(i\) allows us to express the square roots of negative numbers. When you see \(\sqrt{-x}\) for some positive number \(x\), you can rewrite this as \(\sqrt{x}i\). This is because \(i^2 = -1\), making \(i\) a bridge from real to imaginary number calculations.
The imaginary unit is essential for transforming and simplifying operations involving square roots of negative numbers. In problems such as \(\sqrt{-72}\) or \(\sqrt{-6}\), replace the negative sign with \(i\), giving \(\sqrt{72}i\) and \(\sqrt{6}i\). This is the key in expressing calculations that result in imaginary numbers.
- Example: Convert \(\sqrt{-9}\) to \(3i\) since \(\sqrt{-9} = \sqrt{9}i = 3i\).
Square Roots
Square roots are mathematical operations used to find a number which, when multiplied by itself, yields the original number. For example, the square root of 9 is 3, because \(3 imes 3 = 9\).
However, when dealing with negative numbers, square roots lead us to complex numbers using the imaginary unit \(i\). Consider \(\sqrt{-72}\); it can be split into \(\sqrt{72} \times \sqrt{-1}\), which simplifies to \(\sqrt{72}i\).
This splitting process makes complex number calculations possible. Understanding how to handle both positive and negative numbers under square roots is crucial. In the case of positives, you find the factor pairs that simplify the product. For negatives, always insert \(i\) into your calculations.
This simplifies computations and assists with operations such as division or multiplication of square roots within fractions. Understanding these simplifications is important.
However, when dealing with negative numbers, square roots lead us to complex numbers using the imaginary unit \(i\). Consider \(\sqrt{-72}\); it can be split into \(\sqrt{72} \times \sqrt{-1}\), which simplifies to \(\sqrt{72}i\).
This splitting process makes complex number calculations possible. Understanding how to handle both positive and negative numbers under square roots is crucial. In the case of positives, you find the factor pairs that simplify the product. For negatives, always insert \(i\) into your calculations.
This simplifies computations and assists with operations such as division or multiplication of square roots within fractions. Understanding these simplifications is important.
- Example: Simplify \(\sqrt{-50}\) into \(\sqrt{50}i\), and then further by factorizing 50 to \(5\times10\), becoming \(\sqrt{25\times2}i = 5\sqrt{2}i\).
Simplification Steps
Simplification is the process of making a mathematical expression easier to understand or solve. During simplification, your aim is to reduce fractions, combine like terms, or express numbers in their most straightforward form. For example, consider the fraction \(\frac{\sqrt{-72}}{\sqrt{-6}}\).
Firstly, express each square root in terms of \(i\). You get \(\frac{\sqrt{72}i}{\sqrt{6}i}\). Naturally, you can then cancel out the \(i\) terms. You're left with \(\frac{\sqrt{72}}{\sqrt{6}}\).
Next, simplify the square root fraction itself: \(\sqrt{\frac{72}{6}} = \sqrt{12}\). Continue simplifying by factorizing 12 into its prime factors, \(4\times3\), and recognize that \(\sqrt{12} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). Thus, \(\frac{\sqrt{-72}}{\sqrt{-6}}\) simplifies to \(2\sqrt{3}\).
It's essential to perform these operations step by step.
Firstly, express each square root in terms of \(i\). You get \(\frac{\sqrt{72}i}{\sqrt{6}i}\). Naturally, you can then cancel out the \(i\) terms. You're left with \(\frac{\sqrt{72}}{\sqrt{6}}\).
Next, simplify the square root fraction itself: \(\sqrt{\frac{72}{6}} = \sqrt{12}\). Continue simplifying by factorizing 12 into its prime factors, \(4\times3\), and recognize that \(\sqrt{12} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). Thus, \(\frac{\sqrt{-72}}{\sqrt{-6}}\) simplifies to \(2\sqrt{3}\).
It's essential to perform these operations step by step.
- Cancel out common variables or terms.
- Simplify square roots using factorization.
- Rewrite final expressions in their simplest form.
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