Problem 24 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ -\sqrt{-24} $$

Step-by-Step Solution

Answer

\(-\sqrt{-24} = -2i\sqrt{6}\).

Step 1: Understand the Problem

We need to express \(-\sqrt{-24}\) in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\). This involves removing the negative expression inside the square root by using \(i\).

Step 2: Simplify the Negative Inside the Square Root

Since we have \(-\sqrt{-24}\), let's first simplify \(-24\) by expressing it as a product of \(-1\) and \(24\): \(-24 = (-1) \times 24\).

Step 3: Rewrite Using the Imaginary Unit

We know \(i = \sqrt{-1}\), so \(\sqrt{-24} = \sqrt{(-1) \times 24}\). This can be rewritten as \(\sqrt{-1} \times \sqrt{24}\), which is \(i \times \sqrt{24}\).

Step 4: Simplify \(\sqrt{24}\)

The number \(24\) can be factored into prime numbers to simplify its square root: \(24 = 2^3 \times 3\). The square root of \(24\) can be simplified by taking out the factor of a perfect square: \(\sqrt{24} = \sqrt{2^2 \times 2 \times 3} = 2 \times \sqrt{6}\).

Step 5: Combine Simplified Terms

Substitute back our simplified \(\sqrt{24}\) into \(i \times \sqrt{24}\): \(i \times 2 \times \sqrt{6} = 2i\sqrt{6}\).

Step 6: Apply the Original Negative Sign

Remember the original expression was \(-\sqrt{-24}\). So, the expression becomes \(- (2i\sqrt{6}) = -2i\sqrt{6}\).

Key Concepts

Complex Numbers

Complex numbers are vital in mathematics, especially when dealing with negative square roots. They are composed of a real part and an imaginary part. The imaginary part involves the imaginary unit, denoted by \(i\). Here, \(i\) is defined as \(\sqrt{-1}\). This means any negative square root can be rewritten using \(i\). For instance, if you encounter \(\sqrt{-24}\), a part of its simplification is expressing it in terms of \(i\) because \(-24\) is negative.

In general, a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers. In our example, the complex expression \(-2i\sqrt{6}\) is purely imaginary because its real part \(a\) is zero.

• The real component can be zero, making the number purely imaginary.
• Both real and imaginary components can be non-zero, giving a full complex number like \(3 + 4i\).

Complex numbers extend the real numbers, allowing for the square roots of negative numbers to exist.

Square Roots

Square roots are mathematical expressions that find a number whose square equals the original number. For example, the square root of 25 is 5 since \(5^2 = 25\).

Handling square roots of negative numbers can be tricky unless we venture into the realm of complex numbers.

• Ordinarily, \(\sqrt{x}\) implies a non-negative number if \(x\) is non-negative.
• For negative numbers, like \(-24\), it's not possible to find a real number whose square is negative.

Consequently, mathematicians introduced the imaginary unit \(i\) to make sense of these operations. By setting \(i = \sqrt{-1}\), square roots of negative numbers become manageable: \(\sqrt{-1} = i\) thus \(\sqrt{-24} = i\sqrt{24}\). Here, the negative is dealt with by using \(i\).

Using square roots in this way allows for a wider range of algebraic manipulations and solutions.

Simplifying Radicals

Simplifying radicals is about reducing expressions under a square root sign to their simplest form. When simplifying, you're seeking perfect square factors.

For example, given \(\sqrt{24}\), the prime factorization reveals \(24 = 2^3 \times 3\). A perfect square like \(2^2\) can be factored out and simplified: \(\sqrt{24} = \sqrt{2^2 \times 2 \times 3} = 2\sqrt{6}\).

• Identify perfect squares in the factorization step, such as \(4, 9, 16\), etc.
• Take the square root of these perfect squares outside of the radical.

In our solution, this process helped simplify \(\sqrt{24}\) to \(2\sqrt{6}\), which was crucial for expressing \(\sqrt{-24}\) in the term \(-2i\sqrt{6}\). Mastering radical simplification simplifies the work in algebra, particularly in conjunction with complex numbers.