Problem 62 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 5^{\frac{3}{2}} $$

Step-by-Step Solution

Answer

The expression \(5^{\frac{3}{2}}\) simplifies to \(5\sqrt{5}\).

Step 1: Understand the Exponent

The given expression is \(5^{\frac{3}{2}}\). The exponent \(\frac{3}{2}\) can be broken down into two parts: \(\frac{1}{2}\) which represents a square root, and \(3\) which represents the cube of a number.

Step 2: Convert to Radical Expression

Recognize that \(5^{\frac{3}{2}}\) can be rewritten as \((5^{1/2})^3\). Here, \(5^{1/2}\) indicates the square root of \(5\). Hence, we write it as \(\sqrt{5}\).

Step 3: Simplify the Expression

Now, write the expression as a radical with exponentiation. The expression becomes \((\sqrt{5})^3\), which means the cube of the square root of \(5\). This is expressed in radical notation as \(\sqrt{5^3}\).

Step 4: Final Form and Simplify

Calculate \(5^3\), which equals \(125\), and then express the answer as \(\sqrt{125}\). To simplify \(\sqrt{125}\), note that \(125 = 5^3 = 5 \times 5 \times 5\). Thus, \(\sqrt{125} = \sqrt{5^2 \cdot 5} = 5 \sqrt{5}\).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that showcases the idea of repeated multiplication. For instance, when you see something like \(5^3\), it means you multiply 5 by itself three times: \(5 \times 5 \times 5\). However, exponents aren't just limited to whole numbers. They can also be fractions, which is where things get interesting and useful in more complex math problems. In the expression \(5^{\frac{3}{2}}\), the exponent \(\frac{3}{2}\) can be thought of as a combination of operations.

• The denominator \(2\) indicates square root operation.
• The numerator \(3\) indicates cubing, or raising to the power of 3.

Understanding these dual roles of fractional exponents is crucial. It allows us to simplify expressions by turning them into radical forms that can be easier to work with.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest possible form. This process makes computations easier and solutions clearer. Let's use the radical expression \(\sqrt{125}\) as an example. Here’s how we simplify it:

• First, factorize 125 to find perfect squares. We get: \(125 = 5 \times 25\) or \(125 = 5^3\), where \(5^2\) is a perfect square.
• Write \(\sqrt{125}\) as \(\sqrt{5^2 \times 5}\).
• Utilize the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Apply this to get: \(\sqrt{5^2} \times \sqrt{5}\).
• Since \(\sqrt{5^2} = 5\), the expression simplifies to: \(5 \sqrt{5}\).

Simplifying radicals is all about recognizing patterns and breaking them down into smaller, manageable parts. This helps not just in academic exercises, but in practical calculations as well.

Square Roots

Square roots are a specific type of radical where the index is 2. This means you're looking for a number that, when multiplied by itself, gives the original number back. For example, in simple cases, \(\sqrt{16} = 4\) since \(4 \times 4 = 16\). When you're dealing with expressions like \(5^{\frac{1}{2}}\), you're naturally thinking about the square root.Keep these points in mind:

• Simplifying a square root often involves identifying perfect squares within the number.
• Knowing common perfect squares, like 1, 4, 9, 16, etc., can make simplification easier.

To better understand, imagine simplifying \(\sqrt{5^3}\) which ultimately became \(\sqrt{125}\). Factoring 125 as \(5^2 \times 5\), emphasizes the importance of square root applications, showing how perfectly squares play a pivotal role in simplifying radical expressions.