Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$81^{3 / 2}$$

Step-by-Step Solution

Answer

The value of \( 81^{3/2} \) is 729.

Step 1: Understand the Expression

The expression given is \( 81^{3/2} \). This expression is a power with a fractional exponent, which can also be interpreted using the rules of exponents and radicals.

Step 2: Break Down the Fractional Exponent

A fractional exponent like \( 3/2 \) can be translated into a radical. Specifically, \( x^{m/n} = \sqrt[n]{x^m} \). Thus, \( 81^{3/2} \) can be rewritten as \( (\sqrt{81})^3 \).

Step 3: Calculate the Square Root

First, find the square root of 81. The square root of 81 is 9, since \( 9^2 = 81 \). Therefore, \( \sqrt{81} = 9 \).

Step 4: Raise the Result to the Power

Now, use the result from Step 3 and raise it to the 3rd power: \( 9^3 = 9 \times 9 \times 9 \).

Step 5: Compute the Final Multiplication

Calculate \( 9 \times 9 = 81 \), then multiply 81 by 9 again to get 729. Thus, \( 9^3 = 729 \).

Step 6: Conclusion

The value of \( 81^{3/2} \) is 729.

Key Concepts

Radicals

Radicals are essentially the opposite of exponents and are used to find root values of numbers. When dealing with radicals, the symbol \(\sqrt{}\) is used to denote the root being calculated. For example, the square root of a number asks what number multiplied by itself results in the given number. In the expression \(81^{3/2}\), the process involves taking the square root of 81 before proceeding to other operations. Radicals can also be expressed in terms of fractional exponents, where a radical like \(\sqrt[n]{x}\) can be written as \(x^{1/n}\).
Understanding how radicals and exponents relate allows for conversion between the two, which is essential in simplifying and calculating expressions efficiently. In this exercise, recognizing that \(81^{1/2}\) is equivalent to \(\sqrt{81}\) shows the power of interpreting radicals in mathematical expressions.

Exponent Rules

Exponent rules provide a framework for handling operations involving powers. They offer ways to simplify expressions where numbers are raised to various powers. When dealing with fractional exponents, such as in the expression \(81^{3/2}\), it's important to understand these rules. The key rule applied here is \(x^{m/n} = \sqrt[n]{x^m}\), which helps to convert a fractional power into a radical expression.

• Product of powers: \(x^a \times x^b = x^{a+b}\)
• Power of a power: \((x^a)^b = x^{a \times b}\)
• Fractional exponents: Convert to radicals, e.g., \(x^{1/n} = \sqrt[n]{x}\)

In this example, understanding how \(81^{3/2}\) can be broken down into \((\sqrt{81})^3\) is crucial. These transformations simplify computations and make complex exponential evaluations more manageable.

Square Root

A square root is a number which, when multiplied by itself, returns the original number. It is a specific example of a radical where the root degree is 2. In mathematical notation, the square root of a number \(x\) is represented as \(\sqrt{x}\). In the solution, finding the square root of 81 is straightforward: you're looking for a number that, when squared, equals 81. This number is 9, since \(9 \times 9 = 81\).
Understanding square roots helps simplify expressions with fractional exponents. For instance, recognizing that \(81^{1/2}\) is the same as \(\sqrt{81}\) allows for a more straightforward calculation in the given problem.

Exponential Expressions

Exponential expressions involve numbers raised to a power and are fundamental in representing large and small values compactly. They are written in the form \(x^n\), where \(x\) is the base and \(n\) is the exponent. In the problem \(81^{3/2}\), the expression describes raising 81 to the power of \(3/2\). This involves two operations:

• Finding the square root of the base: \(81^{1/2}\)
• Exponentiating the result: \(9^3\)

This step-by-step operation transforms complex tasks into manageable ones. The problem simplifies first by calculating \(9 = \sqrt{81}\) and then by computing \(9^3 = 9 \times 9 \times 9\) to get 729. Understanding how to manipulate exponential expressions, especially with fractional powers, is a useful skill in both pure and applied mathematics.