Problem 21 - Precalculus Student Solutions Manual 5th

Question

Write each number as the product of a real number and i. $$\sqrt{-288}$$

Step-by-Step Solution

Answer

12i \sqrt{2}

Step 1: Identify the imaginary unit

Recognize that \( \sqrt{-288} \) contains a negative number under the square root, which indicates the presence of the imaginary unit \(i\). Recall that \(i = \sqrt{-1}\).

Step 2: Factor the negative sign out of the square root

Rewrite \( \sqrt{-288} \) as \( \sqrt{-1 \cdot 288} \). By the properties of square roots, this can be separated into \( \sqrt{-1} \cdot \sqrt{288}\).

Step 3: Simplify using the imaginary unit

Since \( \sqrt{-1} = i \), we can rewrite the expression as \( i \cdot \sqrt{288}\).

Step 4: Simplify the square root of the positive number

Find the square root of 288. Rewrite 288 as the product of its factors: \( 288 = 144 \cdot 2 \). Since \( \sqrt{144} = 12\), we have \( \sqrt{288} = \sqrt{144 \cdot 2} = 12 \cdot \sqrt{2}\).

Step 5: Combine all parts

Combine the simplified results: \( i \cdot \sqrt{288} = i \cdot (12 \cdot \sqrt{2}) = 12i \sqrt{2}\).

Key Concepts

imaginary unit

When dealing with the square root of a negative number, we enter the realm of complex numbers, specifically the imaginary unit. In mathematics, the imaginary unit is denoted by the symbol \( i \), defined as \( i = \sqrt{-1} \). This concept is crucial for simplifying expressions that involve square roots of negative numbers.

• For example, \(\sqrt{-4} \) can be rewritten as \( \sqrt{4 \times -1} \), which is \( 2i \) because \( \sqrt{4} = 2 \) and \( \sqrt{-1} = i \).
• This applies generally, such that any \(\sqrt{negative} \) can be expressed in terms of \( i \).

The imaginary unit allows mathematicians and students to work within a well-defined system of complex numbers, making calculations involving square roots of negative numbers much more manageable.

square root

The square root function is a fundamental mathematical operation. It involves finding a number which, when multiplied by itself (squared), gives the original number. For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

• When the square root is applied to a negative number, it incorporates the imaginary unit \( i \).
• In the case of \(\sqrt{-288} \), the process first splits the negative part: \( \sqrt{288 \times -1} \).

This separation allows us to write the expression as \( \sqrt{288} \sqrt{-1} \). Now, knowing \( \sqrt{-1} = i \), we can split the initial complex square root into manageable parts.

simplification

Simplification is the process of reducing an expression or equation to its simplest form. When simplifying \( \sqrt{-288} \), follow these steps:

• First, factor out the negative unit as described: \( \sqrt{-288} = \sqrt{288 \times -1} \).
• Then, use the property \( \sqrt{ab} = \sqrt{a} \sqrt{b} \). This gives us \( \sqrt{288} \sqrt{-1} \).
• Next, since \( \sqrt{-1} = i \), we get \( i \sqrt{288} \).
• Finally, break down \( \sqrt{288} \). By factoring 288 into 144 and 2, we can find \( \sqrt{144 \times 2} = 12 \sqrt{2} \).

Combining everything together, we're left with \( 12i \sqrt{2} \). Through these steps, an initially complex expression becomes a much simpler form, easier to understand and use in further calculations.