Problem 27 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ 4^{-3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{64}\).

Step 1: Understanding Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive of that exponent. For example, if you have an expression of the form \(a^{-b}\), it can be rewritten as \(\frac{1}{a^b}\). With this in mind, let's apply this concept to the expression \(4^{-3}\).

Step 2: Rewriting with Positive Exponent

Applying the rule from the previous step, rewrite \(4^{-3}\) as \(\frac{1}{4^3}\). This changes the problem into finding the value of \(4^3\) and then taking the reciprocal.

Step 3: Evaluating the Exponent

Now calculate \(4^3\). To do so, multiply 4 by itself three times: \(4 \times 4 \times 4 = 64\).

Step 4: Finding the Reciprocal

Since \(4^3 = 64\), substitute back into the expression \(\frac{1}{4^3}\) to get \(\frac{1}{64}\).

Step 5: Verify with Calculator

Use a calculator to compute \(4^{-3}\). You should confirm that the result matches the same value, \(\frac{1}{64}\), which ensures our hand calculations are correct.

Key Concepts

Reciprocal

When dealing with negative exponents, understanding the concept of reciprocal is crucial. The reciprocal of a number is simply one divided by that number. For example, the reciprocal of 2 is 1/2. When we encounter a negative exponent, like in the expression \(4^{-3}\), it signals that we need to take the reciprocal of the base raised to the positive exponent.

This means that \(4^{-3}\) is equivalent to \(\frac{1}{4^3}\). The negative exponent "flips" the base, creating a fraction with 1 as the numerator and the base's result as the denominator, making our example turn into \(\frac{1}{64}\) when fully computed.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, which is the exponent. In simple terms, the exponent tells us how many times to multiply the base by itself.

In our exercise, we had to compute \(4^3\), which means multiplying 4 by itself three times: \(4 \times 4 \times 4 = 64\). Thus, the expression \(4^{-3}\) can be rewritten as \(\frac{1}{64}\) when considering its reciprocal.

When you are solving problems with exponents, remember that the operation can significantly increase the value of numbers very quickly. Always ensure each multiplication step is accurate to avoid errors.

Calculator Verification

After evaluating an expression by hand, it's a great idea to verify it with a calculator to ensure accuracy. For negative exponent problems like \(4^{-3}\), this involves checking that our calculated result of \(\frac{1}{64}\) is indeed correct.

Using a calculator, enter \(4^{-3}\). Most scientific calculators will have a direct input for negative exponents, allowing you to quickly verify that the calculated hand result matches the electronic computation.

• Enter the base (4),
• Use the exponent function (often marked as \(x^y\) or similar),
• Input the negative exponent (-3),
• Confirm the result matches your manual calculation.

This process helps catch any errors that might have occurred during manual calculations and provides confidence in your solution.