Problem 66 - Algebra 2

Question

PREREQUISITE SKILL Evaluate each expression. $$ \frac{1-2^{7}}{1-2} $$

Step-by-Step Solution

Answer

The value of the expression is 127.

Step 1: Evaluate the Exponentiation

First, evaluate the part of the expression that involves exponentiation, which is \(2^7\). Calculate \(2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128\).

Step 2: Simplify the Numerator

Substitute the value of \(2^7\) back into the expression. The numerator is \(1 - 2^7\), which simplifies to \(1 - 128 = -127\).

Step 3: Evaluate the Denominator

Next, evaluate the denominator \(1 - 2\). This simplifies directly to \(1 - 2 = -1\).

Step 4: Complete the Division

Now, divide the simplified numerator by the simplified denominator: \(-127 \div -1 = 127\).

Key Concepts

Exponentiation

Exponentiation is the mathematical process where a number is multiplied by itself a certain number of times. This is usually expressed in the form of a base and an exponent, such as in the expression \(2^7\). Here, the base is 2, and the exponent is 7, indicating that 2 should be multiplied by itself 7 times.
The calculation is as follows:

• Start with 2
• 2 x 2 = 4
• 4 x 2 = 8
• 8 x 2 = 16
• 16 x 2 = 32
• 32 x 2 = 64
• 64 x 2 = 128

Thus, the expression \(2^7\) equals 128. Knowing how to handle exponentiation is crucial for solving expressions that involve powers.

Numerator Simplification

Once exponentiation is solved, you can simplify the numerator. The numerator of an expression is the top part of a fraction over the denominator. In our exercise, the numerator is expressed as \(1 - 2^7\).
Here's how you simplify it:

• Evaluate the exponentiation first: \(2^7 = 128\)
• Substitute 128 into the expression: \(1 - 128\)
• Perform the subtraction: \(1 - 128 = -127\)

The simplified numerator becomes \(-127\). It's important to combine like terms and perform arithmetic operations to reduce expressions into simpler forms.

Division Operations

Division operations are a key step after both the numerator and the denominator have been evaluated. Division is the process of determining how many times one number, the divisor, can be subtracted from another, the dividend, until you reach zero. In our exercise, the division is of the form \(\frac{-127}{-1}\).
Consider the steps:

• The dividend is -127 (the result from the numerator).
• The divisor is -1 (the result from the denominator).
• Perform the division: dividing \(-127\) by \(-1\) simplifies to 127, because a negative number divided by a negative number results in a positive number.

Understanding division with sign rules helps ensure accuracy when simplifying expressions.

Denominator Evaluation

Denominator evaluation comes before division to make sure the fraction is ready to be simplified. The denominator is the number below the fraction line, which tells you into how many parts the whole is divided. In the exercise, the denominator is \(1 - 2\).
Let's simplify it:

• You directly subtract: \(1 - 2 = -1\)

Now that we have -1 as the denominator, it's crucial to remember how it affects division. The sign of the denominator impacts the final result of the division. A negative denominator turns the entire division into a potential positive if the numerator is also negative. Evaluating the denominator correctly sets the stage for successful calculation of the entire expression.