Problem 14 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression evaluates to 5.

Step 1: Understanding Negative Exponents

When we have a negative exponent, it means we take the reciprocal of the base and then raise it to the positive of that exponent. For example, for any nonzero number \(a\) and any integer \(n\), \(a^{-n} = \frac{1}{a^n}\). So, \( \left(\frac{1}{3}\right)^{-2} = 9 \) and \( \left(\frac{1}{2}\right)^{-2} = 4 \).

Step 2: Evaluate the First Term

Using the rule for negative exponents, evaluate \( \left(\frac{1}{3}\right)^{-2} \). This gives: \( \left(\frac{1}{3}\right)^{-2} = 3^2 = 9 \).

Step 3: Evaluate the Second Term

Similarly, evaluate \( \left(\frac{1}{2}\right)^{-2} \). This results in: \( \left(\frac{1}{2}\right)^{-2} = 2^2 = 4 \).

Step 4: Subtract the Values

Now subtract the second term from the first: \( 9 - 4 \).

Step 5: Find the Final Answer

Calculate the subtraction: \( 9 - 4 = 5 \).

Key Concepts

Exponent Rules

Exponent rules, especially when dealing with negative exponents, can initially seem daunting, but they become much simpler with practice.
Understanding these rules is crucial for evaluating expressions efficiently. Let's break down how to handle negative exponents:
* The key idea behind a negative exponent is the reciprocal.
* If you see a negative exponent, it tells you to flip the base to its reciprocal.
* Then raise this reciprocal to the absolute value of the exponent.
For example, consider an expression like \( \left(\frac{1}{3}\right)^{-2} \). Here, the base \( \frac{1}{3} \) flips to \( 3 \), and then it's squared because the exponent is \(-2\). This means:\[\left(\frac{1}{3}\right)^{-2} = 3^2 = 9\] Similarly, for \( \left(\frac{1}{2}\right)^{-2} \), the base \( \frac{1}{2} \) becomes \(2\), and it's raised to the power of 2 due to the negative exponent:\[\left(\frac{1}{2}\right)^{-2} = 2^2 = 4\] These rules of negative exponents apply universally to any nonzero base and integer exponent.

Reciprocals

Reciprocals play an essential role in understanding negative exponents. If you grasp how reciprocals work, managing negative exponents will seem much easier.
Simply put, the reciprocal of a number is what you multiply it by to get 1.
For instance:

• The reciprocal of \(3\) is \(\frac{1}{3}\)
• The reciprocal of \(\frac{1}{3}\) is \(3\)

When dealing with expressions such as \( \left(\frac{1}{3}\right)^{-2} \), the negative exponent indicates that you should take the reciprocal:
The fraction becomes its reciprocal, and the negative exponent becomes a positive one. That’s why \( \left(\frac{1}{3}\right)^{-2} \) turns into \(3^2\), or 9. Take \( \left(\frac{1}{2}\right)^{-2} \), and the process is the same: take the reciprocal of \(\frac{1}{2}\), which is \(2\), and square it to get 4.
Understanding reciprocals provides a solid foundation for working with negative exponents and their applications in arithmetic and algebra.

Simplifying Expressions

Simplifying expressions involving exponents can sometimes feel complex, yet it becomes significantly more manageable by applying consistent methods. When you simplify expressions with negative exponents, follow these straightforward steps:
* Convert any negative exponents to positive by using reciprocals.
* Compute any exponential values, essentially squaring or raising the number to the given power.In our example with \( \left(\frac{1}{3}\right)^{-2} - \left(\frac{1}{2}\right)^{-2}\), we handled each term individually first.
By dealing with \( \left(\frac{1}{3}\right)^{-2} \) and \(\left(\frac{1}{2}\right)^{-2}\), we performed a simplification for each term:* Convert \( \left(\frac{1}{3}\right)^{-2} \) to \(3^2 = 9\)* Convert \( \left(\frac{1}{2}\right)^{-2} \) to \(2^2 = 4\)Finally, to obtain the answer, subtract the simplified second term from the first:\[9 - 4 = 5\]Successfully simplifying expressions means tackling one part at a time, ensuring clarity and ease in the calculation process.