Problem 8 - Elementary and Intermediate Algebra

Question

Complete each table. \(\begin{array}{|c|c|}\hline x & {(-9)^{x}} \\ \hline 2 & {} \\ \hline 1 & {} \\\ \hline 0 & {} \\ \hline-1 & {} \\ \hline-2 & {} \\ \hline\end{array}\)

Step-by-Step Solution

Answer

- When x = 2, (-9)^x = 81. - When x = 1, (-9)^x = -9. - When x = 0, (-9)^x = 1. - When x = -1, (-9)^x = -\frac{1}{9}. - When x = -2, (-9)^x = \frac{1}{81}.

Step 1: Understanding Exponentiation

The expression (-9)^x represents a number -9 raised to the power of x. We need to find the result of this expression for different values of x.

Step 1: Calculate (-9)^x when x = 2

Raise -9 to the power of 2: \[ (-9)^2 = (-9) imes (-9) = 81 \] The value of the expression when x = 2 is 81.

Step 2: Calculate (-9)^x when x = 1

Raise -9 to the power of 1: \[ (-9)^1 = -9 \] The value of the expression when x = 1 is -9.

Step 3: Calculate (-9)^x when x = 0

Any number raised to the power of 0 is 1:\[ (-9)^0 = 1 \]Thus, the value when x = 0 is 1.

Step 4: Calculate (-9)^x when x = -1

Raise -9 to the power of -1, which is the reciprocal of the number:\[ (-9)^{-1} = \frac{1}{-9} = -\frac{1}{9} \]The value when x = -1 is -\frac{1}{9}.

Step 5: Calculate (-9)^x when x = -2

Raise -9 to the power of -2, which is the reciprocal of the square:\[ (-9)^{-2} = \frac{1}{(-9)^2} = \frac{1}{81} \]The value when x = -2 is \frac{1}{81}.

Key Concepts

Negative Bases

When dealing with exponentiation, negative bases can be tricky. A base is the number that is multiplied by itself, while an exponent tells us how many times to multiply the base. For negative bases, the pattern of results depends on whether the exponent is even or odd.
For example, with the base

• When the exponent is even, the result is positive. This is because multiplying two negative numbers results in a positive number, so with an even exponent, negatives are paired off. For instance, \((-9)^2 = 81\).
• When the exponent is odd, the result remains negative, since an extra negative remains unpaired. For example, \((-9)^1 = -9\).

Understanding these patterns helps in predicting the outcome when working with negative bases.

Reciprocals

Reciprocals are a concept we often encounter when raising numbers to negative exponents. The reciprocal of a number is essentially 1 divided by that number. In the case of exponentiation:

• Raising a number to the power of -1 is like flipping it, i.e., taking its reciprocal. For example, \((-9)^{-1} = \frac{1}{-9} = -\frac{1}{9}\).
• When we have a negative exponent like -2, it indicates that we take the reciprocal of the number squared, such as \((-9)^{-2} = \frac{1}{(-9)^2} = \frac{1}{81}\).

Using reciprocals helps us compute negative exponents easily, transforming what might seem complex into a simpler division problem.

Zero Exponent Rule

The zero exponent rule is a fundamental principle in mathematics that states any non-zero number raised to the power of zero equals one. This might seem strange at first, but it ensures consistency across rules and simplifies calculations. Here's the breakdown:

• This rule applies regardless of whether the base number is positive or negative. Therefore, both \(9^0 = 1\) and \((-9)^0 = 1\) are true.
• Understanding why this works involves examining the pattern of dividing numbers by themselves as we decrease exponents. This reduces down to the notion that any number divided by itself equals one.

The zero exponent rule is a handy shortcut in algebra, allowing computations to avoid cumbersome multiplications.