Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$27^{4 / 3}$$

Step-by-Step Solution

Answer

The value of \( 27^{4/3} \) is 81.

Step 1: Understand the Expression

The expression to evaluate is \(27^{4/3}\), which involves taking a power with a fractional exponent.

Step 2: Break Down the Fractional Exponent

The expression \( a^{m/n} \) can be interpreted as \( (a^m)^{1/n} \) or \( (a^{1/n})^m \). Here, we have \( n = 3 \) and \( m = 4 \).

Step 3: Evaluate the Cube Root

Calculate \( 27^{1/3} \). The cube root of 27 is 3, because \( 3^3 = 27 \).

Step 4: Raise to the Fourth Power

Now, raise the result from Step 3 to the fourth power: \( 3^4 = 81 \).

Step 5: Conclude the Evaluation

The expression \( 27^{4/3} \) evaluates to 81.

Key Concepts

Rational Exponents

Rational exponents, also known as fractional exponents, are a way to represent expressions of roots and powers. When you see an exponent in the form of a fraction, such as \( a^{m/n} \), it tells you that the base \( a \) is being raised to a power \( m \) and also taken to a root \( n \). This dual action combines raising to a power and taking a root in one compact notation.

Here's how you can interpret \( a^{m/n} \):

• \( a^{m/n} = (a^{1/n})^m \) – This means you first take the \( n^{th} \) root of \( a \), and then raise the result to the power of \( m \).
• Another way to express it is \( a^{m/n} = (a^m)^{1/n} \), where you first raise \( a \) to the power of \( m \), and then take the \( n^{th} \) root of the result.

This flexibility allows you to approach evaluating such expressions in a way that might be easier or more straightforward, depending on the numbers involved. It's particularly useful when simplifying complex exponential expressions.

Exponent Rules

Exponent rules are essential for manipulating expressions with powers and can simplify calculations significantly. These rules apply to both integer and rational exponents. Knowing these rules makes it easier to work through problems like \( 27^{4/3} \).

Here are some key exponent rules to remember:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \) – When multiplying like bases, add the exponents.
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \) – When dividing like bases, subtract the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{mn} \) – When raising a power to another power, multiply the exponents.
• Power of a Product Rule: \( (ab)^n = a^n \times b^n \) – Distribute the exponent to each factor in a product.
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \) – A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.

By applying these rules, you can simplify and solve expressions more effectively. In our example, taking \((27^{1/3})^4\) effectively applies the power of a power rule.

Evaluating Expressions

Evaluating expressions with rational exponents involves methodically applying the rules and definitions you've learned. Let's take the example \( 27^{4/3} \) to understand the process.

First, recognize the fraction \( \frac{4}{3} \), which instructs us to both root and power the base number, depending on how you tackle it. It can be helpful to break it into manageable steps:

• Cubic Root: Identify \( 27^{1/3} \). The cubic root of 27 is 3, because \( 3 \times 3 \times 3 = 27 \).
• Exponentiation: Take this result, 3, and raise it to the power of 4, resulting in \( 3^4 = 81 \).

By systematically breaking down the expression, such as computing \( 27^{1/3} \) before raising it to the fourth power, you ensure accuracy and understanding. Using rational exponents turns what could be complex calculations into a more straightforward process.