Problem 125 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. Simplify: \(16^{\frac{3}{2}}\)

Step-by-Step Solution

Answer

The simplified version of \(16^{\frac{3}{2}}\) is 64.

Step 1: Understanding Fractional Powers

A fractional power is a way of expressing an exponent as a fraction. It's a short form for representing roots. For instance, \(a^{\frac{1}{n}}\) is the same as the n-th root of a, and \(a^{\frac{m}{n}}\) is the n-th root of a to the m-th power. Here, \(16^{\frac{3}{2}}\) means the square root of 16 raised to the 3. This can be written as \((16^{\frac{1}{2}})^{3}\).

Step 2: Calculating Square Root

The next step is to calculate the square root of 16, which is represented by \(16^{\frac{1}{2}}\). The square root of 16 is 4, so the expression simplifies to \(4^{3}\).

Step 3: Raising to Power

The final step is to calculate the cubic of 4, which is \(4^{3}\). In basic mathematics, this is 4 multiplied by itself thrice. So, \(4^3 = 4 \times 4 \times 4 = 64\)

Key Concepts

Roots and Powers

Fractional exponents provide a concise way to represent roots and powers together. Consider the expression \(16^{\frac{3}{2}}\). Here, the base is 16, and the fractional exponent \(\frac{3}{2}\) signifies a combination of both a root and a power.

In general, a fractional exponent can be broken down as \(a^{\frac{m}{n}}\), which implies taking the n-th root of \(a\) and then raising the result to the m-th power.

• In our example, \(16^{\frac{1}{2}}\) signifies the square root of 16.
• Once we compute this root, we raise the result to the power of 3, as shown by \((16^{\frac{1}{2}})^3\).

This method simplifies complex operations into two manageable steps, aligning roots and powers in one expression.

Exponent Rules

Understanding the rules of exponents is crucial when simplifying expressions involving fractional powers. Exponent rules help form the bridge between different forms and operations with powers.

Consider the rule which states that \(a^{m/n} = (a^{1/n})^m\). This illustrates that any fractional exponent reflects two actions: taking a root and then raising to a power.

• This specific rule emphasizes the order: first interpret the root and then apply the power.
• Another useful rule is \((a^m)^n = a^{m\cdot n}\), which allows for combining powers efficiently.

Leveraging these rules can drastically simplify complex expressions, as we've seen in the example with \(16^{\frac{3}{2}}\). Here, interpreting and applying these exponent rules converts an intimidating expression into a straightforward calculation.

Simplification of Expressions

Simplification involves breaking down expressions into their simplest forms. This process often includes using both arithmetic operations and exponent rules to achieve a clearer, more manageable form.

For example, in simplifying \(16^{\frac{3}{2}}\), the task is to work through the series of steps

• Calculate the square root of 16 to get 4.
• Then, raise 4 to the power of 3, giving 64.

The key to simplification is understanding each part of the expression. Starting from recognizing the fractional exponent, progressing through smaller steps, and finally arriving at a single number.

In many cases, simplifying expressions can make solving math problems less complicated and more intuitive. This strategic approach makes working with fractional exponents accessible for various contexts and problems.