Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ -\sqrt{-72} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-6i\sqrt{2}\).

Step 1: Understand the Problem

The problem asks us to write a radical expression that involves a negative number under the square root in terms of the imaginary unit, \( i \). The imaginary unit is defined as \( i = \sqrt{-1} \).

Step 2: Simplify the Negative Sign

The expression \(-\sqrt{-72}\) involves a negative number inside the square root. We can use the property of the square root of a negative number: \( \sqrt{-a} = \sqrt{a} \times i \). So, we rewrite \( -\sqrt{-72} \) as \(- \times i \times \sqrt{72} \).

Step 3: Simplify \(\sqrt{72}\)

To simplify \(\sqrt{72}\), factor 72 into its prime factors to find the perfect squares: \(72 = 2^3 \times 3^2\). The perfect square here is \(3^2 = 9\). So \(\sqrt{72} = \sqrt{9 \times 8} = \sqrt{9} \times \sqrt{8} = 3 \times \sqrt{8}\).

Step 4: Simplify \(\sqrt{8}\)

Simplify further by factoring 8 into its prime factors: \(8 = 2^3\). The perfect square here is \(2^2 = 4\), so \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \times \sqrt{2}\).

Step 5: Combine Simplifications

Combine the simplifications from Steps 3 and 4: \(\sqrt{72} = 3 \times 2 \times \sqrt{2} = 6\sqrt{2}\).

Step 6: Substitute Back

Substitute back into the equation from Step 2: \(- \times i \times \sqrt{72} = -i \times 6\sqrt{2}\).

Step 7: Final Expression

This simplification yields the final result: \(-6i\sqrt{2}\).

Key Concepts

Imaginary Unit

When dealing with the square root of a negative number, the concept of the imaginary unit, denoted as \(i\), comes into play. This unit is a fundamental building block in mathematics for extending the real number system. By definition, \(i\) is equal to the square root of negative one, or \(i = \sqrt{-1}\). This allows us to handle expressions like \(\sqrt{-a}\) by rewriting them in terms of \(i\). Basically, the expression \(\sqrt{-a}\) can be simplified to \(i\sqrt{a}\), allowing us to separate the negative sign from the rest of the radicand.
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Using \(i\) is essential for simplifying expressions such as \(-\sqrt{-72}\). Instead of being stuck with a scary-looking negative inside a square root, we can transform it into a multiple of \(i\). This conversion plays a crucial role in expressing complex numbers, as it helps combine both real and imaginary parts clearly.

Simplifying Radicals

Simplifying radicals involves reducing a square root expression to its simplest form. To do this, one must identify and extract perfect squares from under the radical. Let's take the expression \( \sqrt{72} \) as an example. We start by factoring 72 into its prime factors: \( 72 = 2^3 \times 3^2 \).

Among these, \(3^2\) is a perfect square, which can be easily extracted from under the square root. Therefore, \( \sqrt{72} \) can be simplified:

• \( \sqrt{72} = \sqrt{9 \times 8} \)
• \( \sqrt{72} = \sqrt{9} \times \sqrt{8} \)
• \( \sqrt{72} = 3 \times \sqrt{8} \)

Further simplifying \( \sqrt{8} \):

• The prime factorization gives \( 8 = 2^3 \), and \(2^2\) is a perfect square.
• Thus, \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \times \sqrt{2} \).

Now, multiplying all components ends up with \(\sqrt{72} = 6\sqrt{2}\). Breaking down radicals into simpler components makes calculations less cumbersome and easier to handle.

Perfect Squares

Perfect squares play a pivotal role in simplifying radical expressions. A perfect square is a number that is the product of an integer multiplied by itself, like 1, 4, 9, 16, and so on. Recognizing perfect squares within larger numbers allows us to simplify square roots efficiently.
Consider the number 72. Its factorization reveals a component of 9, which happens to be \(3^2\), a perfect square. This insight helps simplify \(\sqrt{72}\) effectively because the square root of a perfect square, such as 9, converts cleanly to its base, 3 in this case.
The ability to spot and utilize perfect squares is especially beneficial when simplifying expressions in complex numbers, as seen where \(\sqrt{72}\) ultimately simplifies to \(6\sqrt{2}\). By systematically breaking down numbers and spotting these perfect squares, we can make complex calculations simpler and more approachable.