Problem 38 - Algebra for College Students

Question

Change each radical to simplest radical form. \(-\frac{2}{3} \sqrt{96}\)

Step-by-Step Solution

Answer

The simplest form is \\(-\frac{8}{3}\sqrt{6}\\).

Step 1: Simplify the expression inside the radical

First, factor 96 into its prime factors: 96 = 2 × 2 × 2 × 2 × 2 × 3 (or 2^5 × 3). We are looking for perfect squares to simplify. Notice that 2^4 is a perfect square, which is 16.

Step 2: Split the radical

Break down \(\sqrt{96}\) using the perfect square: \(\sqrt{96} = \sqrt{16 \times 6}\).The perfect square \(\sqrt{16}\) can be simplified to 4.

Step 3: Simplify the radical

Use the rule \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to simplify further: \(\sqrt{96} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}\).

Step 4: Distribute the coefficient outside the radical

Multiply the expression \(-\frac{2}{3} \) by the simplified radical: \(-\frac{2}{3} \times 4\sqrt{6} = -\frac{8}{3}\sqrt{6}\).

Key Concepts

Prime Factorization

Prime factorization is a technique used to break down numbers into their basic building blocks, which are prime numbers. This method helps in simplifying various mathematical expressions, including radicals. A prime number is a number that has only two divisors: 1 and itself.
To find the prime factorization of a number, you keep dividing the number by the smallest possible prime until you are left with only prime factors.
Here's how to apply prime factorization using an example with the number 96:

• Divide 96 by 2, which is the smallest prime number. We get 48.
• Continue dividing by 2: 48 ÷ 2 = 24.
• Keep dividing: 24 ÷ 2 = 12.
• We divide again: 12 ÷ 2 = 6.
• Finally, 6 ÷ 2 = 3. The number 3 is a prime, so we stop here.

Now, the prime factorization of 96 is expressed as \(2^5 \times 3\). This breakdown helps in identifying and extracting perfect squares, which are crucial for simplifying radical expressions.

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. Recognizing perfect squares is vital for simplifying radicals, as they can be easily taken out of the square root.
For example, numbers like 1, 4, 9, 16, 25, and 36 are perfect squares because they correspond to \(1^2, 2^2, 3^2, 4^2, 5^2,\) and \(6^2\), respectively.
When simplifying radicals, it simplifies the process significantly if you spot a perfect square within a number.

• For example, in the number 96, the factor \(2^4=16\) is a perfect square.
• This realization allows you to extract \(\sqrt{16}\) from under the radical, simplifying it to 4, as seen in the step-by-step solution.

Understanding and identifying perfect squares allows you to break down complex radical expressions into simpler, more manageable parts.

Radical Expressions

Radical expressions include numbers or expressions within a root symbol, like a square root. Simplifying radical expressions often involves breaking down what's inside the root into easily manageable components.
The simplification process can be made smoother by:

• First, breaking down the number under the radical into its prime factors.
• Identifying any perfect squares among those factors.
• Applying the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), which assists in separating the perfect squares and simplifying the expression.
• In our case, simplifying \(\sqrt{96}\) by splitting it into \(\sqrt{16 \times 6}\) utilizes the identified perfect square 16 to simplify the expression to \(4\sqrt{6}\).
  This separation and simplification reduce complexity and is essential for rendering radical expressions into their simplest forms.