Problem 70 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the expression in simplest radical form. $$ -\sqrt{\frac{8}{3}} $$

Step-by-Step Solution

Answer

The simplified radical form of \(-\sqrt{\frac{8}{3}}\) is \(-\frac{2\sqrt{6}}{3}\).

Step 1: Simplify the numerator

The numerator is 8. We will look for the largest perfect square that is a factor of 8. The most significant perfect square in 8 is 4. We know that 8 = 4 × 2 and 4 is a perfect square because it is equal to \(2^2\). So, we have: \[ -\sqrt{\frac{8}{3}} = -\sqrt{\frac{4 \times 2}{3}} \]

Step 2: Separate the square root

Now, separate the square root of fractions into the square root of the numerator and the square root of the denominator: \[ -\sqrt{\frac{4 \times 2}{3}} = -\frac{\sqrt{4 \times 2}}{\sqrt{3}} \]

Step 3: Simplify the square roots

Simplify the square roots of the numerator and denominator. Since we have the square root of a perfect square \(4 = 2^2\), it simplifies to 2. Note that we can't simplify \(\sqrt{2}\) and \(\sqrt{3}\) as they are both prime numbers. So, we have: \[ -\frac{\sqrt{4 \times 2}}{\sqrt{3}} = -\frac{2\sqrt{2}}{\sqrt{3}} \]

Step 4: Rationalize the denominator

Rationalize the denominator by multiplying both, the numerator, and the denominator by \(\sqrt{3}\): \[ -\frac{2\sqrt{2}}{\sqrt{3}} = -\frac{2\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{6}}{3} \] The expression \(-\sqrt{\frac{8}{3}}\) in simplest radical form is: \(-\frac{2\sqrt{6}}{3}\).

Key Concepts

Perfect Square

Understanding perfect squares can make simplifying square roots much easier. A perfect square is a number that can be expressed as the square of an integer. For example, numbers like 1, 4, 9, 16, and 25 are perfect squares because they can be written as \(1^2, 2^2, 3^2, 4^2, \) and \(5^2\) respectively.

In the step-by-step solution provided, we identified 4 as a perfect square because it is \(2^2\). This realization helped us break down the square root of 8 to find the simplest radical form. Recognizing and using perfect squares can significantly simplify expressions that involve square roots. It allows you to break down complex numbers into simpler components, making the calculations more manageable.

• Always look for the largest perfect square factor when simplifying a square root.
• This can help isolate simple numbers and separate them from irrational parts.

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that the denominator is a rational number. When a denominator contains a square root, like \(\sqrt{3}\) in this case, it is often considered more elegant and standardized to "rationalize" it.

To do this, you can multiply both the numerator and the denominator by a number that will eliminate the square root in the denominator. In our example, multiplying by \(\sqrt{3}\) accomplished this:

• \(-\frac{2\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{6}}{3}\)
• This approach leaves you with a rational denominator (3, in this case).

Rationalizing the denominator simplifies the expression and aligns it with standard mathematical conventions. It makes the expression easier to interpret and work with in subsequent calculations.

Simplifying Square Roots

Simplifying square roots involves expressing a square root in its simplest form. It often means reducing the number inside the square root to its lowest possible terms.

Consider the original square root expression in our problem, \(\sqrt{\frac{8}{3}}\):

1. **Break Down into Factors:** We identified that 8 can be expressed as \(4 \times 2\), where 4 is a perfect square.
2. **Use Perfect Squares:** Understanding that \(4 = 2^2\), we simplified \(\sqrt{4}\) to 2.
3. **Combine What's Left:** The simplification process gave us \(\frac{2\sqrt{2}}{\sqrt{3}}\). This left us with a more straightforward expression to work with.

Simplifying square roots helps strip away complex layers, leading to expressions that are easier to comprehend and manipulate. This skill is essential to solving equations and problems that involve roots, making complex mathematical tasks simpler and more intuitive.