Problem 5 - College Algebra

Question

Evaluate each exponential expression. $$ -2^{6} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the base and the exponent

We identify the base of the exponential expression, which is -2, and the exponent, which is 6.

Step 2: Apply the exponential rule

Using the rules of exponents, we know that a base raised to an even power is positive. So, we repeatedly multiply -2 by itself, 6 times: -2 x -2 x -2 x -2 x -2 x -2

Step 3: Perform the multiplication

Multiplying the factors together, we get 64.

Key Concepts

Base and Exponent

In any exponential expression, understanding the terms "base" and "exponent" is essential. The *base* is the number that is being multiplied. The *exponent* tells us how many times to multiply the base by itself. For example, in the expression \(-2^6\), -2 is the base, and 6 is the exponent.

**Why Does This Matter?**
* Knowing the base guides our multiplication process.* The exponent indicates the number of times this multiplication happens.* Recognizing these elements ensures we follow exponential rules correctly.

Consider the example \(-2^6\):

• The base is -2, implying we'll use this number.
• The exponent is 6, so multiply -2 by itself six times.

Understanding the distinction between base and exponent is fundamental to solving exponential expressions accurately.

Even Power Rule

The even power rule comes into play when dealing with exponential expressions where the exponent is an even number. This rule is special because regardless of whether the base is positive or negative, the result of the multiplication will be positive.

For example, in \(-2^6\):

• The base is negative (-2).
• The exponent is an even number (6).

When we apply the even power rule, we know that multiplying an even number of negative factors results in a positive product. This happens because pairs of negative numbers multiply to positive ones.

**Let's Break It Down:**

• First pair: (-2) \(\times\) (-2) = 4
• Second pair: (-2) \(\times\) (-2) = 4
• Third pair: (-2) \(\times\) (-2) = 4

Now, multiplying these together as \(4 \times 4 \times 4 = 64\),we see the outcome is positive 64. The even power rule simplifies predicting outcomes when dealing with negative bases.

Multiplication of Factors

Now that we identified the base and the exponent, and how the even power rule applies, let's delve into multiplying factors. It's crucial in exponential calculations to perform each multiplication step carefully.

Here's how you handle the multiplication for \(-2^6\):

• Write the base (-2) six times to represent the power of 6.
• Begin multiplying in pairs, keeping the even power rule in mind.
• Combine the results of each multiplication step.

For \(-2\):

• Multiply first: (-2 \(\times\) -2) = 4
• Second multiplication: (-2 \(\times\) -2) = 4
• Third multiplication: (-2 \(\times\) -2) = 4

Finally, multiply these results together: \(4 \times 4 \times 4 = 64\).
Through exact multiplication, we ensure accurate results, demonstrating how all steps connect smoothly from understanding the base and exponent to the final multiplication.