Problem 2 - Algebra for College Students

Question

Simplify each numerical expression. \(2^{-4}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{16}\).

Step 1: Understanding the Exponent

The expression we have is \(2^{-4}\). The negative exponent indicates that we should take the reciprocal of the base, \(2\), and then apply the positive exponent. This means \(2^{-4} = \frac{1}{2^4}\).

Step 2: Compute the Positive Power

Now we need to calculate \(2^4\). This means multiplying the base, \(2\), by itself 4 times: \(2 \times 2 \times 2 \times 2 = 16\).

Step 3: Apply the Reciprocal

Using the result from the previous step, replace \(2^4\) by \(16\) in the reciprocal. It gives us: \(\frac{1}{16}\).

Step 4: Final Expression

The simplified expression for \(2^{-4}\) is thus \(\frac{1}{16}\).

Key Concepts

Negative Exponents

When dealing with exponents, a negative exponent implies that we need to take the reciprocal of the base and then raise it to the absolute value of the exponent. In simpler terms, if you see something like \(a^{-n}\), just remember it equals \(\frac{1}{a^n}\). This tells us that instead of multiplying, we're dividing.
For example, in the problem \(2^{-4}\), the '-4' indicates we flip over the base \(2\) to get \(\frac{1}{2^4}\). We then deal with the positive exponent. Negative exponents often confuse at first, but once you grasp the idea of flipping the base, it becomes straightforward!

• Start by writing the base in the denominator.
• Convert the exponent sign to positive.
• Calculate as normal, but just remember it's now in the form of a fraction.

Reciprocal

In mathematics, the reciprocal of a number is simply \(\frac{1}{\text{that number}}\). It’s all about flipping, and this idea is crucial for understanding negative exponents.
The reciprocal changes the position from the numerator to the denominator or vice versa. For instance, the reciprocal of 2 is \(\frac{1}{2}\), meaning you're considering one part out of two.
Consider our example:

• The initial expression is \(2^{-4}\).
• By taking its reciprocal, we transform it to \(\frac{1}{2^4}\).

By understanding this concept, you're better equipped to work with negative exponents and simplify expressions involving them.

Power of a Number

Understanding the power of a number is key to grasping exponents. The power indicates how many times you multiply the base by itself. For instance, \(2^4\) means you multiply 2 by itself four times: \(2 \times 2 \times 2 \times 2\).
The result of such a multiplication is called the 'power'. When solving expressions like \(2^{-4}\), first compute \(2^4\) to get 16.

• The base is the number being multiplied.
• The exponent (or power) tells you the count of the multiplication.
• In \(2^4\), 2 is multiplied four times, resulting in 16.

This concept helps make solving exponent problems less intimidating.

Simplifying Expressions

Simplifying expressions involves breaking them down to their simplest form, making them easier to understand or use. In our example, \(2^{-4}\), simplifying involves working through each step systematically.
Begin with the negative exponent, move to calculating the power, and finally, use the reciprocal.

• Convert negative exponents using reciprocals.
• Calculate the power of the number.
• Combine the results to express the simplest form possible.

For \(2^{-4}\), you simplify it to \(\frac{1}{16}\). This method ensures clarity and helps avoid errors in complex calculations. Step-by-step handling ensures that you fully grasp the transition from a complicated to a clear expression.