Problem 18 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ 5^{-3} $$

Step-by-Step Solution

Answer

The evaluated expression \(5^{-3}\) is \(\frac{1}{125}\).

Step 1: Understand the Expression

The expression given is \(5^{-3}\), which involves an exponent with a negative power. The negative exponent indicates the reciprocal of the base raised to the opposite positive power.

Step 2: Apply the Negative Exponent Rule

According to the negative exponent rule, \(a^{-n} = \frac{1}{a^n}\). Hence, \(5^{-3} = \frac{1}{5^3}\).

Step 3: Calculate the Positive Exponent

Now, calculate \(5^3\):\[5^3 = 5 \times 5 \times 5 = 125\]

Step 4: Find the Reciprocal

The expression becomes \(\frac{1}{125}\) since \(5^{-3} = \frac{1}{5^3}\). This is the final evaluated expression.

Step 5: Verify with a Calculator

Using a calculator, computing \(5^{-3}\) should also give you \(0.008\) or \(\frac{1}{125}\) as the result, confirming our manual calculation.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving a base and an exponent. In the expression \(5^{-3}\), 5 is the base and \(-3\) is the exponent. Exponents can be positive, negative, or even zero, and each has special rules. A positive exponent tells you how many times to multiply the base by itself. However, with a negative exponent, the operation is slightly different. Instead of multiplying, the base is moved into the denominator of a fraction, essentially creating an inverse, or reciprocal, of the base. This is why we convert \(5^{-3}\) to \(\frac{1}{5^3}\). Here, understanding the exponent means grasping how the operation affects the base. A negative exponent always means taking the reciprocal of the base raised to the absolute value of the exponent. So, next time you see a negative exponent, remember you're simply being asked to divide rather than multiply.

Reciprocal

The reciprocal of a number is simply one divided by that number. In the context of exponents, a negative exponent signals that you should take the reciprocal of the base. For example, in the expression \(5^{-3}\), we take the reciprocal of \(5^3\).

• The reciprocal of \(5\) is \(\frac{1}{5}\).
• For \(5^3\), it's \(\frac{1}{5^3}\), or \(\frac{1}{125}\).

This reciprocal represents the same value whether you compute it manually or use a calculator. It's essential to understand that the reciprocal transforms the operation from a multiplication into a division. This concept is crucial, as it forms the basis for evaluating negative exponents accurately. The reciprocal, therefore, is a critical part of understanding how negative exponents function.

Calculator Verification

Once you've performed calculations manually, it's often a good idea to check your work using a calculator. This is called calculator verification. It serves as a useful double-check to make sure you haven't missed a step or made an error in arithmetic. For the expression \(5^{-3}\), inputting this directly into a calculator should return \(0.008\), verifying the steps you've taken by hand.Checking your work with a calculator not only confirms your answer; it can also reinforce your understanding of the process.

• Ensure you've entered the expression properly.
• Look at the calculation output to verify its accuracy with your manual result.
• If there is a discrepancy, it can be a cue to revisit the calculation steps.
• Calculator verification can help illuminate any areas of difficulty.

Using both manual and calculator methods offers a complete approach to solving expressions involving exponentiation, particularly with negative exponents.