Problem 114 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ (16)^{-5 / 4} $$

Step-by-Step Solution

Answer

The expression \((16)^{-5/4}\) simplifies to \(\frac{1}{32}\).

Step 1: Understand the Negative Exponent

Recall that a negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, \((a^{-n}) = \frac{1}{a^n}\). Therefore, \((16)^{-5/4}\) becomes \(\frac{1}{(16)^{5/4}}\).

Step 2: Simplify the Power 4/5

Next, we need to simplify \((16)^{1/4}\), which is the fourth root of 16. Since 16 is equal to \(2^4\), \((16)^{1/4} = (2^4)^{1/4} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2\).

Step 3: Raise the Result to the Power of 5

Now, using the result from Step 2, \((16)^{1/4} = 2\), apply the power to carry out \((16)^{5/4} = (16)^{1/4 \times 5} = 2^5\).

Step 4: Calculate the Final Power

Calculate \(2^5\), which is \(2 \times 2 \times 2 \times 2 \times 2 = 32\). Therefore, \((16)^{5/4} = 32\).

Step 5: Complete the Reciprocal Operation

Substitute back into the expression \((16)^{-5/4}= \frac{1}{32}\) as shown in Step 1.

Key Concepts

Understanding Rational Exponents

Rational exponents can seem confusing at first, but breaking down the concept makes them much easier to understand. A rational exponent represents a combination of a root and a power. It can always be expressed in the form of a fraction, with the numerator showing the power and the denominator indicating the root.
For instance, in the expression \(a^{m/n}\), \(m\) is the power, and \(n\) is the root. This means that you first take the \(n\)-th root of \(a\) (this is the denominator's role), and then raise the result to the power of \(m\) (numerator's role).

• Example: \(16^{5/4}\) signifies taking the fourth root of 16 first, then raising the result by the fifth power.
• This breakdown simplifies calculations by addressing each part of the exponent separately.

The Concept of Reciprocals

The term reciprocal might sound a bit mathematical, but it's a straightforward concept. A reciprocal of a number is just one divided by that number. Mathematically, if you have a number \(a\), its reciprocal is expressed as \(\frac{1}{a}\).
When dealing with negative exponents, as in the expression \(a^{-n}\), a reciprocal plays a major role because the exponent's negativity indicates you need to take the reciprocal.

• Example: \(16^{-5/4}\) transforms to \(\frac{1}{16^{5/4}}\) after applying the reciprocal.
• This method allows us to handle negative exponents by changing the expression's form to operate further towards simplification.

Understanding this concept removes a lot of intimidation from negative exponents!

Simplifying Expressions with Exponents

Simplifying expressions is all about breaking complex equations into more understandable and manageable forms. With exponents, this process often involves manipulating the exponent using rules of arithmetic.

• Steps to Simplify: You typically start by calculating the root indicated by the denominator of your rational exponent. Then, handle any power the expression is raised to.
• For instance, simplifying \(16^{5/4}\) involves initially finding \(16^{1/4}\), which is the fourth root of 16, i.e., 2. Raise this result to the fifth power (2 raised to 5), giving you 32.
• Finally, apply any needed reciprocal if a negative exponent was involved as shown before.

This method breaks complexity into step-by-step actions, aiding in internalizing the process of exponential simplification.