Problem 18 - Algebra 2

Question

Evaluate each expression if \(a=3, b=0.3, c=\frac{1}{3},\) and \(d=-1\). \(\frac{a-10 b}{c^{2} d^{2}}\)

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Substitute Values

First, substitute the given values for each variable in the expression \(\frac{a-10b}{c^{2}d^{2}}\). Here, \(a = 3\), \(b = 0.3\), \(c = \frac{1}{3}\), and \(d = -1\). Therefore, the expression becomes \(\frac{3 - 10(0.3)}{\left(\frac{1}{3}\right)^{2}(-1)^{2}}\).

Step 2: Simplify the Numerator

Calculate the value in the numerator. This means computing \(3 - 10(0.3)\). First, calculate \(10(0.3) = 3\), and then \(3 - 3 = 0\). Thus, the numerator simplifies to \(0\).

Step 3: Simplify the Denominator

Next, simplify the denominator \(\left(\frac{1}{3}\right)^{2}(-1)^{2}\). First, \(\left(\frac{1}{3}\right)^{2} = \frac{1}{9}\), and since \((-1)^{2} = 1\), the denominator becomes \(\frac{1}{9} \times 1 = \frac{1}{9}\).

Step 4: Compute the Final Expression

Now divide the simplified numerator by the simplified denominator. The expression is \(\frac{0}{\frac{1}{9}}\), which equals \(0\) since any number divided by itself is \(0\).

Key Concepts

Substitution

Substitution is a fundamental process in algebra that involves replacing variables in expressions with given numerical values. In our example, we're tasked with evaluating the expression \(\frac{a-10b}{c^{2}d^{2}}\) using the values \(a=3\), \(b=0.3\), \(c=\frac{1}{3}\), and \(d=-1\). To substitute:

• Replace \(a\) with \(3\)
• Replace \(b\) with \(0.3\)
• Replace \(c\) with \(\frac{1}{3}\)
• Replace \(d\) with \(-1\)

After substitution, the expression becomes: \[\frac{3 - 10(0.3)}{\left(\frac{1}{3}\right)^{2}(-1)^{2}}\]This step transforms the algebraic expression into an arithmetic one, making it easier to perform calculations.

Simplification

Simplification is the process of reducing an expression to its simplest form. In this exercise, we have to simplify both the numerator and the denominator after substitution. First, address the numerator:

• We start with \(3 - 10(0.3)\).
• Calculate \(10 \times 0.3 = 3\).
• Subtract: \(3 - 3 = 0\). The numerator simplifies to \(0\).

This is a crucial simplification step, as the result directly impacts the outcome of the entire expression. A similar approach is taken to simplify the denominator.

Numerator and Denominator Operations

In algebraic fractions, it's crucial to handle operations with both the numerator and the denominator. Given the expression \[\frac{0}{\left(\frac{1}{3}\right)^{2}(-1)^{2}}\] focus on simplifying each part separately. **Simplifying the Denominator:**

• Start by calculating \(\left(\frac{1}{3}\right)^{2}\) which equals \(\frac{1}{9}\).
• Then, calculate \((-1)^{2}\) which gives \(1\).
• Thus, the denominator simplifies to \(\frac{1}{9} \times 1 = \frac{1}{9}\).

Handling these operations accurately ensures a correct evaluation of the fraction.

Fraction Division

Fraction Division is one of the key skills in evaluating expressions involving fractions. Once the numerator and denominator are simplified, divide the two. In this example, the expression boils down to \[\frac{0}{\frac{1}{9}}\]When you divide any number by another, remember:

• If the numerator is zero, the division always results in zero.
• The reasoning is \(0 \times 9 = 0\), which confirms our result.

Here, the final answer is clearly \(0\), as dividing zero by any fractional number results in zero. This concept helps immensely in solving complex algebraic expressions with fractions.