Problem 60 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \frac{\sqrt{-96}}{\sqrt{2}} $$

Step-by-Step Solution

Answer

The solution simplifies to \( 4i\sqrt{3} \).

Step 1: Express the square root of a negative number in terms of i

Recognize that we can rewrite \( \sqrt{-96} \) using imaginary unit \( i \) such that \( \sqrt{-96} = \sqrt{96} \times \sqrt{-1} = \sqrt{96}i \). This is because \( i \) is defined as \( \sqrt{-1} \).

Step 2: Simplify the radical expression \( \sqrt{96} \)

The number 96 can be factored into prime numbers as \( 96 = 2^5 \times 3 \). To simplify \( \sqrt{96} \), we take out pairs of twos out of the square root: \( \sqrt{96} = \sqrt{2^5 \times 3} = 2^2 \times \sqrt{6} = 4\sqrt{6} \).

Step 3: Substitute back into the expression with i

Substitute \( \sqrt{96} \) from Step 2 back into \( \sqrt{-96} = \sqrt{96}i \) such that \( \sqrt{-96} = 4\sqrt{6}i \).

Step 4: Perform the division operation

Divide the expression \( \frac{4\sqrt{6}i}{\sqrt{2}} \) as indicated in the problem. Simplify the division: \( \frac{4\sqrt{6}i}{\sqrt{2}} = 4i\frac{\sqrt{6}}{\sqrt{2}} = 4i\sqrt{\frac{6}{2}} = 4i\sqrt{3} \).

Step 5: Simplify the expression

The expression \( \frac{4i \sqrt{6}}{\sqrt{2}} \) simplifies to \( 4i \sqrt{3} \). This is the simplified form of the original expression in terms of \( i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit, \(i\), is defined as \(\sqrt{-1}\). This means any square root of a negative number can be expressed using imaginary numbers. For example, \(\sqrt{-96}\) can be written as \(\sqrt{96}i\).

• Real Part: The component of the complex number that does not involve \(i\).
• Imaginary Part: The component of the complex number that involves \(i\).
• Imaginary Unit: Denoted by \(i\).

Understanding complex numbers helps in solving quadratic equations with no real solutions and in performing operations on radical expressions involving negative numbers.

Square Root Simplification

Simplifying square roots is about making expressions easier to work with. The goal is to take out any perfect square factors from under the square root sign. For example, if you have \(\sqrt{96}\), begin by factoring it into prime factors: \(96 = 2^5 \times 3\). From here, you can identify pairs of factors, since every pair of numbers inside the square root can be brought out as a single number.

Steps to Simplify:

• Factor the number into primes.
• Identify pairs of primes.
• Bring each pair outside the square root as a single integer.

So \(\sqrt{96}\) becomes \(4\sqrt{6}\). This method of simplification makes calculations with square root expressions more manageable.

Radical Expressions

Radical expressions contain a square root, cube root, or any higher roots. When dealing with these, understanding how to manipulate and simplify them is important. Simplifying a radical can involve factoring numbers into their prime components and bringing out multiples as fully simplified terms.

Key Points:

• Can include square roots, cube roots, etc.
• Simplified by removing perfect powers.
• Expressed with the square root symbol, '\(\sqrt{}\)'.

Working with radicals often requires combining like terms, converting complex expressions into simpler forms—like rewriting \(\sqrt{-96}\) as \(4\sqrt{6}i\)—and understanding operations like addition, subtraction, multiplication, and division involving these expressions.

Division of Radicals

Division of radicals involves simplifying expressions where square roots or other radicals are present in the numerator, denominator, or both. Such as \(\frac{\sqrt{-96}}{\sqrt{2}}\), which we simplify by converting into imaginary numbers first and then performing division.

How to Approach:

• Simplify each radical separately before dividing.
• Use the property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).
• Rationalize if necessary to remove radicals from the denominator.

For example, \(\frac{4\sqrt{6}i}{\sqrt{2}}\) simplifies to \(4i\sqrt{3}\) because \(\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{3}\). This process helps ensure the complete expression is as simple as possible.