Problem 13 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$16^{-3 / 4}$$

Step-by-Step Solution

Answer

The value of \( 16^{-3/4} \) is \( \frac{1}{8} \).

Step 1: Identify the Base and Exponent

The expression given is \( 16^{-3/4} \). Here, 16 is the base, and \(-3/4\) is the exponent. The negative exponent indicates that the expression is the reciprocal of the base raised to a positive exponent.

Step 2: Rewriting the Negative Exponent

Rewrite the expression using the negative exponent rule: \( a^{-n} = \frac{1}{a^n} \). Thus, \( 16^{-3/4} = \frac{1}{16^{3/4}} \).

Step 3: Simplify the Expression with Positive Exponent

To simplify \( 16^{3/4} \), use the property that \( a^{m/n} = \sqrt[n]{a^m} \). Here, we have \( 16^{3/4} = (16)^{1/4}^3 \).

Step 4: Evaluate the Fourth Root

Evaluate \( 16^{1/4} \). The fourth root of 16 is the number which raised to the power of 4 equals 16. Since \( 2^4 = 16 \), \( 16^{1/4} = 2 \).

Step 5: Raise to the Power of 3

Now, raise the result from Step 4 to the power of 3: \( 2^3 = 8 \).

Step 6: Determine the Reciprocal

From Step 2, since our exponent is originally negative, take the reciprocal of the result we found: \( \frac{1}{8} \).

Key Concepts

Negative Exponents

In mathematics, we often encounter expressions with negative exponents. A negative exponent signifies that the base should be inverted or turned into its reciprocal, which means flipping it to the bottom of a fraction if it is initially on the top, and vice versa. For instance:

• If you have an expression such as \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \).
• This principle helps simplify expressions by allowing us to switch between negative and positive exponents, which often makes calculations more manageable.

Therefore, when you see an expression like \( 16^{-3/4} \), initially consider it as \( \frac{1}{16^{3/4}} \). This transformation is the crucial step that simplifies the process, as it converts a more complex operation into something that's easier to evaluate.

Rational Exponents

Rational exponents are a type of exponent that involve fractions. Specifically, they describe roots along with the usual power of a number and provide a way to express radicals more compactly. Here's what you need to know:

• When you see an expression like \( a^{m/n} \), it implies two operations. The base \( a \) is first raised to the power of \( m \), and then you take the \( n^{th} \) root of that result, or vice versa.
• This is interpreted as \( \sqrt[n]{a^m} \). You could also see it as doing the nth root first, then raising it to the m-th power.

In our exercise, \( 16^{3/4} \) stands for taking the fourth root of 16 and then cubing the outcome. This dual operation makes rational exponents a versatile tool for simplifying and evaluating expressions without using a calculator.

Roots

Roots are fundamental operations in mathematics, especially when handling rational exponents. They essentially reverse the process of exponentiation.

• The nth root of a number \( a \), written as \( \sqrt[n]{a} \), is the value that, when raised to the nth power, equals \( a \).
• For example, the square root (\( \sqrt{} \)) is the most common root operation, but other roots like the cube root \( \sqrt[3]{} \) and the fourth root \( \sqrt[4]{} \) are just as important in various applications.

In the given problem, we identify \( 16^{1/4} \) as the fourth root of 16, which equals 2, since 2 raised to the fourth power gives us 16. Understanding roots allows you to easily decompose complex expressions and is essential for manipulating equations with rational exponents.