Problem 42
Question
Perform the indicated operations and write the result in standard form. $$ \sqrt{-12}(\sqrt{-4}-\sqrt{2}) $$
Step-by-Step Solution
Verified Answer
The result of the operation is \(-4\sqrt{3}-2\sqrt{6}i\).
1Step 1: Simplify the square roots
Begin by simplifying each square root. \(\sqrt{-12} = \sqrt{12}i = 2\sqrt{3}i\) since \( \sqrt{-1} = i \) and \(\sqrt{12} = 2\sqrt{3}\). \(\sqrt{-4} = \sqrt{4}i = 2i\). \(\sqrt{2}\) remains as \(\sqrt{2}\). Substituting these values into the equation gives \(2\sqrt{3}i(2i-\sqrt{2})\).
2Step 2: Perform the operations
Next, perform the multiplication operation. This yields \(4\sqrt{3}i^{2}-2\sqrt{6}i = -4\sqrt{3}-2\sqrt{6}i \) because \(i^{2}=-1\).
3Step 3: Write the result in standard form
The result in standard form is \(-4\sqrt{3}-2\sqrt{6}i\). In complex numbers, the standard form is a + bi where a is the real part and b is the imaginary part.
Key Concepts
Imaginary UnitSquare RootsStandard FormSimplification of Expressions
Imaginary Unit
Let's explore the concept of the imaginary unit, denoted by the symbol \(i\). The imaginary unit is a mathematical construct that allows us to work with the square roots of negative numbers.
The fundamental property of \(i\) is that the square of \(i\) equals \(-1\), or mathematically, \(i^2 = -1\).
This property enables us to handle calculations involving negative square roots, which cannot be done in the realm of real numbers without engaging the imaginary unit.
For example, if you encounter \(\sqrt{-4}\), this can be rewritten using the imaginary unit as \(2i\), since \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\).
The fundamental property of \(i\) is that the square of \(i\) equals \(-1\), or mathematically, \(i^2 = -1\).
This property enables us to handle calculations involving negative square roots, which cannot be done in the realm of real numbers without engaging the imaginary unit.
For example, if you encounter \(\sqrt{-4}\), this can be rewritten using the imaginary unit as \(2i\), since \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\).
- Imaginary numbers are of the form \(bi\), where \(b\) is a real number.
- They allow for solutions and operations that are not possible with only real numbers.
Square Roots
Square roots are the inverse operation of squaring a number. The primary goal when dealing with square roots in complex numbers is to simplify them.
In standard arithmetic, the square root of a number \(x\) is a value that, when multiplied by itself, returns \(x\). For non-negative numbers, this is straightforward. However, square roots of negative numbers introduce a twist.
This is where the imaginary unit \(i\) becomes crucial. For instance, the square root of \(-12\) is simplified as \(2\sqrt{3}i\). Breaking this down, \(\sqrt{-12} = \sqrt{12} \cdot \sqrt{-1} = 2\sqrt{3}\cdot i\).
In standard arithmetic, the square root of a number \(x\) is a value that, when multiplied by itself, returns \(x\). For non-negative numbers, this is straightforward. However, square roots of negative numbers introduce a twist.
This is where the imaginary unit \(i\) becomes crucial. For instance, the square root of \(-12\) is simplified as \(2\sqrt{3}i\). Breaking this down, \(\sqrt{-12} = \sqrt{12} \cdot \sqrt{-1} = 2\sqrt{3}\cdot i\).
- Always look to express the square root in its simplest form.
- The imaginary unit is used for the square root of negative numbers.
- Utilize prime factorization to simplify square roots if necessary.
Standard Form
In the world of complex numbers, expressing results in standard form is essential for clarity and proper interpretation.
The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
This format allows us to delineate between the different components of a complex number, aiding in both computation and comprehension.
In the exercise provided, after all operations, \(-4\sqrt{3}-2\sqrt{6}i\)is the resulting complex number in its standard form, with \(-4\sqrt{3}\) as the real part and \(-2\sqrt{6}i\) as the imaginary part.
The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
This format allows us to delineate between the different components of a complex number, aiding in both computation and comprehension.
In the exercise provided, after all operations, \(-4\sqrt{3}-2\sqrt{6}i\)is the resulting complex number in its standard form, with \(-4\sqrt{3}\) as the real part and \(-2\sqrt{6}i\) as the imaginary part.
- Always ensure that the complex number is simplified to this form before concluding your calculations.
- This form is pivotal for performing additional operations such as addition, subtraction, or comparison of complex numbers.
Simplification of Expressions
The simplification of expressions in algebra involves reducing them to their simplest or most comprehensible form. It involves breaking down parts and recombining them efficiently.
When working with complex numbers, particularly those involving square roots and the imaginary unit, simplification follows similar principles.
Begin by simplifying individual terms before performing operations between them. For example, in the expression \(\sqrt{-12}(\sqrt{-4} - \sqrt{2})\), it’s crucial to first simplify each square root: \(\sqrt{-12} = 2\sqrt{3}i\) and \(\sqrt{-4} = 2i\).
After simplification, carry out the operations as demonstrated in step-by-step solutions: distribute, multiply, and combine like terms while observing the properties of \(i\).
When working with complex numbers, particularly those involving square roots and the imaginary unit, simplification follows similar principles.
Begin by simplifying individual terms before performing operations between them. For example, in the expression \(\sqrt{-12}(\sqrt{-4} - \sqrt{2})\), it’s crucial to first simplify each square root: \(\sqrt{-12} = 2\sqrt{3}i\) and \(\sqrt{-4} = 2i\).
After simplification, carry out the operations as demonstrated in step-by-step solutions: distribute, multiply, and combine like terms while observing the properties of \(i\).
- Focus on clarity and precision in the simplification process.
- Work systematically to avoid errors, especially when dealing with multiple terms.
- Regularly check each step for correct use of \(i^2 = -1\).
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