Problem 18 - Precalculus Mathematics for Calculus

Question

Evaluate each expression. (a) \(\left(-\frac{2}{3}\right)^{-3}\) (b) \(\left(\frac{3}{2}\right)^{-2} \cdot \frac{9}{16}\) (c) \(\left(\frac{1}{2}\right)^{4} \cdot\left(\frac{5}{2}\right)^{-2}\)

Step-by-Step Solution

Answer

(a) -\frac{27}{8}, (b) \frac{1}{4}, (c) \frac{1}{100}

Step 1: Evaluate Negative Exponent (a)

To evaluate \(\left(-\frac{2}{3}\right)^{-3}\), recognize that a negative exponent indicates that the base is reciprocal, followed by making the exponent positive. Therefore, \(\left(-\frac{2}{3}\right)^{-3} = \left(-\frac{3}{2}\right)^3\). Calculate this by multiplying \(-\frac{3}{2} \times -\frac{3}{2} \times -\frac{3}{2}\). This gives \(-\frac{27}{8}\).

Step 2: Evaluate Negative Exponent and Multiply (b)

First, address the negative exponent for \(\left(\frac{3}{2}\right)^{-2}\) by taking the reciprocal and making the exponent positive: \(\left(\frac{2}{3}\right)^2\). This gives \(\frac{4}{9}\). Now, multiply this result by \(\frac{9}{16}\): \(\frac{4}{9} \times \frac{9}{16} = \frac{4}{16} = \frac{1}{4}\).

Step 3: Multiply Powers (c)

Start by calculating each power: \(\frac{1}{2}^4 = \frac{1}{16}\) and \(\left(\frac{5}{2}\right)^{-2}\) means taking reciprocal and squaring, to give \(\frac{4}{25}\). Now multiply: \(\frac{1}{16} \times \frac{4}{25} = \frac{4}{400} = \frac{1}{100}\).

Key Concepts

Understanding Reciprocals

The reciprocal of a number is simply what you multiply that number by to get 1. In simpler terms, the reciprocal of a fraction is created by swapping the numerator and the denominator. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). This concept is extremely useful when dealing with negative exponents.
When you encounter a negative exponent, such as \( \left( -\frac{2}{3} \right)^{-3} \), you need to take the reciprocal of the base. Here, \( -\frac{2}{3} \) becomes \( -\frac{3}{2} \), and the exponent is changed from negative to positive. Now you can calculate \(-\frac{3}{2} \times -\frac{3}{2} \times -\frac{3}{2} \) to get \(-\frac{27}{8} \).
Using reciprocals in this way simplifies expressions and makes calculations more straightforward, especially when you need to raise fractions to an exponent.

Multiplying Fractions Made Easy

When multiplying fractions, you do not need to find a common denominator. The process is simply a matter of multiplying across the numerators and multiplying across the denominators. For example, if you have \( \frac{4}{9} \times \frac{9}{16} \), you multiply the numerators: \(4 \times 9 = 36\), and the denominators: \(9 \times 16 = 144\). The result is \( \frac{36}{144} \).
To make things even simpler, always check if you can simplify or reduce the fraction further. Here, \( \frac{36}{144} \) simplifies to \( \frac{1}{4} \) by dividing both the numerator and denominator by their greatest common factor, which is 36.
With practice, multiplying fractions becomes a quick and efficient process. Always remember to check the fractions involved to see if you can simplify them before or after you multiply.

The Art of Simplifying Expressions

Simplifying expressions is an essential skill in math to make calculations manageable. It involves reducing fractions, combining like terms, and sometimes factoring out the greatest common terms.
Take the example \( \frac{1}{16} \times \frac{4}{25} \). First, calculate the product by multiplying the numerators together, \(1 \times 4 = 4\), and the denominators, \(16 \times 25 = 400\). This gives \( \frac{4}{400} \).
To simplify, find the greatest common factor of 4 and 400, which is 4. Dividing both the numerator and the denominator by 4, you get \( \frac{1}{100} \).

• Always start by simplifying each part of the expression.
• Look for common factors and use them to reduce.
• Remember that a simpler expression is always easier to work with.

Regular practice will make you more comfortable with the steps and help you identify simplifications faster.