Problem 32 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$\sqrt{-\frac{64}{36}}$$

Step-by-Step Solution

Answer

\[\sqrt{-\frac{64}{36}} = \frac{4}{3}i\]

Step 1: Recognize the Negative Square Root

Identify the negative sign under the square root, which indicates the presence of an imaginary number. Rewrite the expression as: \[\sqrt{-\frac{64}{36}} = \sqrt{-1} \cdot \sqrt{\frac{64}{36}}\] where \(\sqrt{-1} = i\).

Step 2: Simplify the Imaginary Part

Replace \(\sqrt{-1}\) with \(i\):\[i \cdot \sqrt{\frac{64}{36}}\]

Step 3: Simplify the Rational Square Root

Calculate the square root of the fraction separately:\[\sqrt{\frac{64}{36}} = \frac{\sqrt{64}}{\sqrt{36}}\] Compute each square root: \[\frac{\sqrt{64}}{\sqrt{36}} = \frac{8}{6}\] Reduce the fraction to its simplest form: \[\frac{8}{6} = \frac{4}{3}\]

Step 4: Combine the Results

Combine the simplified square root with the imaginary unit:\[i \cdot \frac{4}{3} = \frac{4}{3}i\]

Step 5: Concluding Step: Final Simplified Form

The simplified form of the expression \(\sqrt{-\frac{64}{36}}\) is:\[\frac{4}{3}i\]

Key Concepts

Square Roots

When dealing with square roots, the basic idea is finding a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. However, things get a bit tricky with negative numbers, like in the expression \( \sqrt{-64} \). Normal square roots for negative numbers are not real, which is where imaginary numbers come into play.

To compute the square root of a negative number, we use the concept of the imaginary unit \( i \), where \( i = \sqrt{-1} \). This means \( \sqrt{-64} \) can be rewritten as \( i \sqrt{64} \). This concept allows us to work with numbers in the complex number system, paving the way for more advanced calculations.

Complex Numbers

Complex numbers are an extension of the real number system and are in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. In the solution, we encountered \( \frac{4}{3}i \), which is a purely imaginary number since there is no real part.

Using complex numbers, we can solve equations that do not have real solutions. The understanding of complex numbers is crucial in mathematics, particularly in fields like engineering and physics, where they help describe wave patterns and electrical currents. The process of expressing terms with \( i \) allows us to handle negative square roots more easily.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form. This is achieved by dividing both the numerator and the denominator by the greatest common divisor. For example, in the step where \( \sqrt{\frac{64}{36}} \) is simplified to \( \frac{8}{6} \), further simplification reduces it to \( \frac{4}{3} \) by dividing both 8 and 6 by 2.

Math problems often require this type of simplification for clarity and correctness. Approaching each component—like the numerator and denominator of a fraction—can make more complex calculations manageable. Simplification ensures that the final answer is not only accurate but also presented in an easy-to-understand format.