Problem 16 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left[\left(\frac{2}{5}\right)^{-2}\right]^{-1} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{4}{25} \).

Step 1: Apply the Exponent Rule

The expression is \( \left(\frac{2}{5}\right)^{-2} \), and it has an exponent of -2. Recall that if you have a negative exponent, you can reciprocate the base to make the exponent positive: \[ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} \]. Therefore, \[ \left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^2 \].

Step 2: Simplify the New Exponent

Now, you have the expression \( \left(\frac{5}{2}\right)^2 \), which means multiplying \( \frac{5}{2} \) by itself: \[\left(\frac{5}{2}\right) \times \left(\frac{5}{2}\right) = \frac{5 \times 5}{2 \times 2} = \frac{25}{4} \].

Step 3: Invert the Expression Again

The entire expression is \( \left[\left(\frac{2}{5}\right)^{-2}\right]^{-1} \). Now you have simplified it to \( \frac{25}{4} \). Since this entire expression also has a negative exponent, \( -1 \), reciprocate the fraction: \[ \left(\frac{25}{4}\right)^{-1} = \frac{4}{25} \].

Key Concepts

Negative Exponents

Exponentiation is a powerful mathematical operation, and understanding negative exponents is a key part of it. When you see a negative exponent, it indicates a reciprocal. Essentially, a negative exponent tells you to "flip" the base.
For instance, if you have an expression like \[ a^{-n} \] this is equivalent to \[ \frac{1}{a^n} \].
This not only changes the position of the base in terms of numerator or denominator, but also converts the exponent to a positive value.
Think of negative exponents as telling you to take the "reciprocal effect" of a number raised to a positive power. The rules apply similarly to fractions, as seen in the original problem. For example, \[ \left(\frac{2}{5}\right)^{-n} \] becomes \[ \left(\frac{5}{2}\right)^{n} \].
Remember:

• When you change a negative exponent to a positive, reciprocate the base.
• Negative exponent rules apply to numerical bases, fractions, and variables alike.
• Always express your final result with positive exponents unless otherwise specified.

Fraction Operations

Fractions are special numbers that represent parts of a whole. To handle fractions correctly in mathematical expressions, it's important to understand the basic operations.
When we operate with fractions, we follow these steps:

• **Addition/Subtraction**: Fractions need a common denominator to add or subtract. Find a common ground and then combine.
• **Multiplication**: Simply multiply the numerators to each other and the denominators to each other.
• **Division**: Multiply by the reciprocal of the divisor.

In exponentiation, especially when dealing with fractions, it's all about manipulating the numerator and the denominator separately. In the given problem, after applying the negative exponent rule, we multiplied \[ \left(\frac{5}{2}\right) \] by itself. This resulted in \[ \frac{25}{4} \].
Multiplying fractions requires you to:

• Keep the work neat and track both parts together for accuracy.
• Simplify fractions at the end for clarity.

Reciprocals

Reciprocals are fundamental in mathematics. They are especially crucial when dealing with division and negative exponents. The reciprocal of a number is essentially "1 divided by that number".
Here’s how it works:

• For a simple integer or a whole number like 2, the reciprocal is \[ \frac{1}{2} \].
• For a fraction, say \[ \frac{a}{b} \], the reciprocal is \[ \frac{b}{a} \].
• Reciprocals effectively flip the number or fraction upside down.
• When applied to a fraction with an exponent, the exponent remains unchanged in value, but becomes positive if it was negative before reciprocation.

In the original exercise, after calculating the initial exponent operation and obtaining \[ \frac{25}{4} \], we needed to further reciprocate because the expression was raised to \[ -1 \]. This flips it to \[ \frac{4}{25} \].
Finding reciprocals can be a quick and straightforward operation that, when used effectively, makes complex mathematical tasks simpler.