Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ \left(-\frac{2}{3}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression without using exponents is \( \frac{16}{81} \).

Step 1: Raise the Fraction to the Fourth Power

First, we need to raise the given expression to the power of 4: \[ \left(-\frac{2}{3}\right)^4 = \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \cdot \left(-\frac{2}{3}\right) \]

Step 2: Multiply the Fractions Together

Now, let's multiply the fractions together. When multiplying fractions, we simply multiply the numerators together and the denominators together. Also, remember that a negative number raised to an even power becomes positive: \[ =(-1)^4\left(\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3}\right) \] \[ = (1)\left(\frac{2\cdot 2\cdot 2\cdot 2}{3\cdot 3\cdot 3\cdot 3}\right) \]

Step 3: Simplify the Fraction

Simplify the fraction by multiplying the numerators and denominators: \[ = \frac{2^4}{3^4} \] \[ = \frac{16}{81} \] Since the fraction is simplified, the final answer is: \[ \boxed{\frac{16}{81}} \]

Key Concepts

Simplifying Fractions

When solving mathematical problems, we often encounter fractions that can be simplified. Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator share no common factors except 1. To do this, find the greatest common factor (GCF) of the numerator and the denominator and divide both by this number.

Remember, fractions can also often be simplified before multiplying them together by canceling out any common factors from the numerator and denominator across the fractions involved. However, in the case of \( (-\frac{2}{3})^4 \), we first need to raise the fraction to the power before simplifying.

Multiplying Fractions

When it comes to multiplying fractions, the process is straightforward. You multiply the numerators together and multiply the denominators together. No common denominator is needed, which is unlike addition or subtraction of fractions.

Let's break it down: if you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is \( \frac{a \cdot c}{b \cdot d} \). In our example, \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \) simplifies to \( \frac{2^4}{3^4} \) by multiplying the numerators and denominators separately.

Negative Exponents

Negative exponents can seem confusing at first, but they follow a simple rule: a negative exponent indicates that the base should be taken as the reciprocal and then raised to the absolute value of the exponent. Essentially, for any non-zero number 'a' and integer 'n', \( a^{-n} = \frac{1}{a^n} \).

In our exercise, the negative sign is tied to the base of the exponent, and not to the exponent itself. This means we don't have to apply the rule for negative exponents directly, since the exponent is actually positive. However, understanding how negative exponents work is crucial for more complex calculations.

Even Powers

Whenever we raise a number to an even power, the result is always non-negative, no matter if the base was positive or negative. This happens because the product of two negative numbers is positive, which translates to our fraction: raising \( -\frac{2}{3} \) to the fourth power (an even number) results in a positive outcome.

Specifically, \( (-1)^{4} = 1 \) and \( (-\frac{2}{3})^{4} = \frac{2^4}{3^4} \) because the negative base \( -\frac{2}{3} \) is raised to the power of 4, resulting in \( \frac{16}{81} \), which is positive. An understanding of even powers can simplify the process of working with exponents and help in predicting the sign of the outcome.