Problem 18 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$16^{-0.25}$$

Step-by-Step Solution

Answer

The value of \(16^{-0.25}\) is \(\frac{1}{2}\).

Step 1: Understand Negative Exponent

The negative exponent in an expression like \(a^{-b}\) means that this is equivalent to the reciprocal of \(a^b\). Therefore, \(16^{-0.25}\) can be rewritten as \(\frac{1}{16^{0.25}}\).

Step 2: Simplify the Fractional Exponent

Next, recognize that a fractional exponent is another way of expressing the root. The expression \(16^{0.25}\) converts to the fourth root of 16, since \(0.25 = \frac{1}{4}\). That means \(16^{0.25} = \sqrt[4]{16}\).

Step 3: Solve the Root

Determine the fourth root of 16. We know that \(2^4 = 16\), therefore \(\sqrt[4]{16} = 2\).

Step 4: Apply Reciprocal from Negative Exponent

Substitute the value from Step 3 back into the reciprocal expression from Step 1. This means \(16^{-0.25} = \frac{1}{2}\).

Key Concepts

Negative Exponents

Negative exponents can initially seem confusing, but once you understand the rules, they become quite manageable. When you encounter a negative exponent, it is a signal to take the reciprocal of the base raised to the corresponding positive exponent. For example, if you see an expression like \(a^{-b}\), it means you should rewrite it as \(\frac{1}{a^b}\). This turns the negative exponent into a positive one.

• Negative exponents are the opposite of positive exponents.
• They indicate division rather than multiplying the base repeatedly.
• Focus on first converting the negative exponent to a positive by taking the reciprocal of the base.

By transforming an expression using these rules, it becomes simpler to calculate, as you would only need to deal with positive exponents thereafter.

Fractional Exponents

Fractional exponents are another interesting aspect of exponents. They provide an effective method for describing roots. When you see an expression like \(a^{\frac{m}{n}}\), it suggests that you are dealing with the \(n\)-th root of a number raised to the \(m\)-th power. This might sound complicated at first, but it can be broken down easily:

• The numerator of the fraction corresponds to the power.
• The denominator signifies the root.

So for \(16^{0.25}\):
- The numerator (1) means the number will remain unchanged in terms of power.- The denominator (4) signifies the fourth root of 16, leading to the expression \(\sqrt[4]{16}\).Working with fractional exponents is a flexible way of processing roots, and understanding this concept allows you to manipulate numbers efficiently.

Roots

Roots help us understand expressions with fractional exponents in a clear geometric manner. The root of a number \(x\) is another number which, when multiplied by itself a specific number of times (depending on the type of root), gives \(x\). With the expression \(\sqrt[n]{x}\), you find a number that multiplies itself \(n\) times to return to \(x\). Here are key points about roots:

• The square root \(\sqrt{x}\) is the number that multiplies by itself to produce \(x\).
• The cube root \(\sqrt[3]{x}\) is similarly defined, but for three instances of multiplication.
• And so on for fourth roots, fifth roots, etc.

For the exercise in question, \(\sqrt[4]{16}\) means finding a number which, when multiplied together four times, results in 16. Since \(2^4 = 16\), the fourth root of 16 is indeed 2. Understanding roots allows you to tackle any problem involving fractional exponents with confidence.