Problem 70 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{(6-5)^{8}-1}{(-9)(-3)-4} $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Simplify the Numerator

The numerator of the expression is \((6-5)^8 - 1\). Begin by calculating \(6-5\), which equals \(1\). Next, compute \(1^8\), which equals \(1\). Finally, subtract \(1\) from \(1\) to get \(0\). So the numerator simplifies to \(0\).

Step 2: Simplify the Denominator

The denominator of the expression is \((-9)(-3) - 4\). First, multiply \(-9\) by \(-3\) to get \(27\). Then, subtract \(4\) from \(27\) to obtain \(23\). So the denominator simplifies to \(23\).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now that we have simplified the numerator to \(0\) and the denominator to \(23\), we can divide the two values. The expression is \(\frac{0}{23}\), which equals \(0\).

Key Concepts

Numerator and Denominator

In the world of fractions, understanding the terms 'numerator' and 'denominator' is crucial. The numerator is the top part of the fraction, which represents how many parts of the whole are being considered. The denominator, on the other hand, is the bottom part and it names the total number of equal parts in the whole. In our original exercise, the expression \[ \frac{(6-5)^{8}-1}{(-9)(-3)-4} \] uses these concepts. Here,

• The entire expression on the top, \((6-5)^8 - 1\), is the numerator. It tells us what is being calculated or focused on.
• The expression in the bottom, \((-9)(-3)-4\), is the denominator. It describes the partitioning or base value involved in the division.
Focusing on simplifying the numerator and the denominator first will prepare us for determining the expression's overall value.

Simplification

Simplification is a process of making an expression easier to understand and solve. For algebraic expressions, this often means reducing them to their simplest form. Take the numerator of our example: \[ (6-5)^8 - 1 \]We simplify the expression step by step:

• Calculate \(6-5\), which equals \(1\). Since any number raised to the power of \(8\) that is \(1\) is still \(1\), the expression becomes \(1^8\) or simply \(1\).
• Next, subtract \(1\) from \(1\), resulting in \(0\).

Now, the denominator: \[ (-9)(-3) - 4 \]Steps for simplification:

• Multiply \(-9\) by \(-3\) to get \(27\). Remember, multiplying two negative numbers results in a positive number.
• Subtract \(4\) from \(27\), which simplifies to \(23\).

Through simplification, we've efficiently broken down potentially complex calculations into straightforward arithmetic.

Division

Division is the operation of sharing or distributing a quantity into equal parts. It is one of the fundamental operations in arithmetic and algebra. In the context of fractions, a division is represented by a numerator and a denominator. When you simplify both parts of a fraction, as in our exercise, you are preparing it for division. In our expression, the division occurs as follows:\[ \frac{0}{23} \]

• Having simplified the numerator to \(0\) and the denominator to \(23\), we perform the division, \(0 \div 23\).
• Dividing zero by any positive non-zero number always results in zero.

This step solidifies our understanding that when the numerator is zero, the value of the entire fraction is zero, regardless of the value of the denominator, as long as it is not zero. Thus, the outcome of our expression evaluation is \(0\).