Problem 90 - College Algebra Essentials

Question

In Exercises \(83-90\), evaluate each expression without using a calculator. $$16^{-\frac{5}{2}}$$

Step-by-Step Solution

Answer

The result of \(16^{-\frac{5}{2}}\) is \(\frac{1}{1024}\).

Step 1: Identify the Base and the Exponent

The base here is 16 and the exponent is -\(\frac{5}{2}\). The negative sign signifies that the reciprocals will be taken and the fraction indicates that the square root of the base will be taken then raised to the 5th power.

Step 2: Simplify the Expression

First ignore the negative sign and focus on the \(\frac{5}{2}\). This fraction can be seen as indicating the square root (due to the 2 in the denominator), raised to the power of 5 (due to the 5 in the numerator). Hence \(16^{\frac{5}{2}}\) is the same as \((\sqrt{16})^5\). Calculate \(\sqrt{16}\) which equals to 4 then raise it to the power of 5 i.e.\((4)^5\) which equals to 1024.

Step 3: Deal with the Negative Exponent

Now that we've taken care of the fraction part of the exponent, turn to the negative part. A negative exponent means that we take the reciprocal of the base. Hence, \(16^{-\frac{5}{2}}\) equals to \(\frac{1}{16^{5/2}}\) or \(\frac{1}{1024}\).

Key Concepts

Understanding Negative Exponents

Exponents tell us how many times to use the base in a multiplication. But what happens when the exponent is negative? This might initially seem confusing, but it follows a simple rule.
Negative exponents indicate that you need to take the reciprocal of the base raised to the absolute value of the exponent. For example, if we have an expression such as \(16^{-\frac{5}{2}}\), the negative sign of the exponent tells us that we need the reciprocal.

• The negative sign means "put it under 1". So, instead of multiplying by 16, we divide by it when raised to the fractional exponent.

Understanding this concept can help in simplifying complex expressions with negative exponents.

Breaking Down Fractional Exponents

Fractional exponents represent both roots and powers. When you see a fractional exponent like \(\frac{5}{2}\), you should split it into two parts: the numerator and the denominator.
The denominator of the fraction indicates the root you are taking, while the numerator tells you which power to raise it to after.
For instance:

• In our expression \(16^{\frac{5}{2}}\), the denominator 2 signifies the square root.
• The numerator 5 indicates that we then raise the result of the square root to the power of 5.

So, start by finding the root, then apply the power for simplifying the expression involving fractional exponents.

The Concept of a Reciprocal

A reciprocal is what you multiply a number by to get 1. In fractions, finding a reciprocal is simple: you flip the fraction upside down. When dealing with whole numbers or even complex expressions, finding the reciprocal means placing the expression under 1.
For example, with a negative exponent like \(16^{-\frac{5}{2}}\), we took the reciprocal in the third step of the solution:

• First, calculate \(16^{\frac{5}{2}}\)
• Then, take the reciprocal: \(\frac{1}{16^{\frac{5}{2}}}\)

Reciprocal transformations simplify the expressions involving negative exponents.

Simplifying Expressions with Exponents

Simplifying expressions with exponents often involves applying multiple rules of exponents. Let's simplify an expression like \(16^{-\frac{5}{2}}\).

• First, tackle the fractional exponent: find the square root of 16, giving 4, and then raise the result to the power of 5, resulting in 1024.
• Next, address the negative exponent by taking the reciprocal, which results in \(\frac{1}{1024}\).

Combining these steps helps untangle complex exponential expressions, making evaluation straightforward. Always start by handling the fractional part before moving on to the reciprocal.