Problem 55 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ 16^{1 / 2} $$

Step-by-Step Solution

Answer

The value of \(16^{1/2}\) is 4.

Step 1: Understanding the Problem

The expression given is \(16^{1/2}\), which is equivalent to finding the square root of 16. In fractional exponents, \(a^{1/n}\) is the same as \(\sqrt[n]{a}\), so here we are looking for \(\sqrt{16}\).

Step 2: Calculating the Square Root

To calculate \(\sqrt{16}\), think of a number that when squared equals 16. The number that satisfies this condition is 4, because \(4^2 = 16\).

Step 3: Approximate the Answer if Needed

The exercise asks to approximate the answer to the nearest hundredth if appropriate. However, since 4 is an integer, no approximation is necessary.

Key Concepts

Fractional Exponents

Fractional exponents are a way to express roots using exponents. Instead of writing the square root symbol \( \sqrt{} \), we use a fraction as the exponent. This method not only makes the expression look more like algebraic equations but also makes it easier to handle in calculations. For example, the expression \( 16^{1/2} \) uses a fractional exponent, where "1/2" represents the square root.

• The number "1" in the fraction tells us that the base number (16) is raised to the power of 1, meaning it stays as 16.
• The "2" in the denominator tells us that we are finding the square root of the base number.

In general, \( a^{m/n} \) is equivalent to \( \sqrt[n]{a^m} \), which means take the \( n \)-th root of \( a \) and then raise the result to the power of \( m \). This notation is particularly useful for solving equations and simplifying expressions that involve roots.

Evaluating Expressions

When it comes to evaluating expressions with fractional exponents, it's all about simplifying them step-by-step. The expression \( 16^{1/2} \) is a straightforward example. Here’s how you can think about evaluating such expressions:

• Recognize the base number and the fractional exponent. In this case, the base is 16 and the exponent is 1/2.
• Convert the fractional exponent into a root. For \( 16^{1/2} \), this means finding the square root of 16.
• Solve the root. The square root of 16 is 4, because \( 4 \times 4 = 16 \).
• If required, round the answer to the nearest hundredth. In this example, rounding isn't needed because the result \( 4 \) is an integer.

By breaking down the process into these steps, you can tackle more complex problems that involve fractional exponents with confidence.

Radical Expressions

Radical expressions involve the use of roots, such as square roots, cube roots, and so on. These expressions are commonly written using the radical symbol \( \sqrt{} \), but they also can be written using fractional exponents.
Square roots are one of the most common types of radical expressions. To work with them:

• Identify the number under the radical sign. For instance, in \( \sqrt{16} \), the number is 16.
• Find a number which when multiplied by itself gives the number under the radical sign. Here, since \( 4 \times 4 = 16 \), the square root of 16 is 4.

Transforming radical expressions into fractional exponents makes it easier to apply algebraic rules. For example, multiplication and division rules for exponents can help simplify radical expressions further. By mastering both forms, you enrich your math toolbox for solving varied mathematical problems.