Problem 3 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$ (-2)^{6} $$

Step-by-Step Solution

Answer

The result of the expression \(-2^6\) is 64

Step 1: Understand the Exponentiation Rule

An exponent refers to the number of times a number is multiplied by itself. For example, \(a^n\) means a is multiplied by itself n times. If n is 6, it means a is multiplied by itself 6 times.

Step 2: Applying the Exponent to -2

In the given expression \(-2^6\), -2 is the base while 6 is the exponent. This implies that -2 is multiplied by itself 6 times.

Step 3: Calculate the Expression

Start multiplying. Note that a negative number multiplied by a negative number gives a positive result.\n\((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2)= 64\)

Key Concepts

Exponentiation Rule

The exponentiation rule is a fundamental concept in mathematics that allows us to simplify how we express repeated multiplication of the same number. When you see any expression in the form of \(a^n\), it's telling us that the base \(a\) is multiplied by itself \(n\) times. For example, \(a^3 = a \times a \times a\). This rule helps us understand and calculate powers more efficiently.

• The base is the number that is being multiplied.
• The exponent shows how many times the base is multiplied by itself.
• If the exponent is positive, like 6 in \((-2)^6\), we keep multiplying.

Understanding and applying this rule correctly ensures accurate calculations, which is crucial, especially when dealing with complex equations. It abstracts repetitive tasks into a simple notation, making mathematical expressions easier to handle.

Negative Bases in Exponents

Handling negative bases in exponents requires a careful approach. When using negative numbers as bases with exponential expressions, the outcome can differ based on whether the exponent is odd or even.Consider the expression \((-2)^6\). Here, -2 is the base, which means:

• If the exponent is even (like 6 in this case), the result becomes positive. This occurs because the pairs of negative numbers multiply to produce positive results, such as \((-2) \times (-2) = 4\).
• If the exponent is odd, the result remains negative as there will be an unpaired negative factor.

In our example, \((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2)\) ultimately resolves to a positive 64, because the six negative factors pair up and cancel out the negativity.

Multiplying Negative Numbers

Multiplying negative numbers is a key aspect of understanding exponential expressions with negative bases. Specific rules apply that must be memorized to accurately solve problems:

• When you multiply two negative numbers, the product is positive: \((-a) \times (-b) = ab\). This happens because the negatives cancel each other out.
• If you multiply a negative number by a positive number, the result is negative: \((-a) \times b = -ab\).
• The repeated product of an even number of negative numbers results in a positive outcome.
• An odd number of negative numbers multiplied together results in a negative product.

Practicing multiplication with negative numbers helps reinforce these rules, making complex expressions more manageable. In exponential expressions like \((-2)^6\), knowing that pairs of negative factors produce positive results provides clarity in computations.