Problem 196 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{6^{2}-4^{2}}{4-6} $$

Step-by-Step Solution

Answer

-10

Step 1 - Identify the expression

We are given the expression to simplify: \[\frac{6^{2}-4^{2}}{4-6}\].

Step 2 - Calculate the exponents

Calculate the squares of 6 and 4 separately: \[6^2 = 36\] and \[4^2 = 16\].

Step 3 - Substitute the values

Substitute the calculated values back into the expression: \[\frac{36 - 16}{4 - 6}\].

Step 4 - Simplify the numerator

Simplify the numerator, which is the difference of squares: \[36 - 16 = 20\].

Step 5 - Simplify the denominator

Simplify the denominator: \[4 - 6 = -2\].

Step 6 - Divide the simplified numerator by the simplified denominator

Divide the results from Step 4 by Step 5: \[\frac{20}{-2} = -10\].

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The exponent tells us how many times the base is multiplied by itself.
For instance, in the expression \(6^2\), 6 is the base and 2 is the exponent, which means that 6 is multiplied by itself, resulting in 36.
Here are some key points to remember about exponentiation:

• Any number raised to the power of 1 remains the same (e.g., \(a^1 = a\)).
• Any number raised to the power of 0 is always 1 (e.g., \(a^0 = 1\), provided \(a eq 0\)).

Understanding exponentiation helps simplify expressions because it allows us to break down complex terms into simpler components.
In our problem, we calculated \(6^2\) and \(4^2\) separately to get 36 and 16 respectively.

Difference of Squares

The difference of squares is a specific type of algebraic expression that takes the form \(a^2 - b^2\). This expression can be factored into \((a + b)(a - b)\).
This factorization property is helpful for simplification, as it breaks down complex polynomials into simpler binomials.
For example, in our problem, the expression \(6^2 - 4^2\) can be seen as the difference of squares where \(a = 6\) and \(b = 4\). Here are the steps to factorize:

• Identify a and b from \(a^2 - b^2\).
• Rewrite the expression as \((a + b)(a - b)\).

Applying this to our problem, we don't directly use the factoring form, but understanding it helps us simplify the numerator:
\((6 + 4)(6 - 4) = 10 \times 2 = 20\).

Division of Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\).
When dividing rational numbers, we apply the rule of simplifying both the numerator and the denominator.
In our exercise, we simplified \(\frac{36 - 16}{4 - 6}\) by first handling the operations separately:

• Simplify the numerator: \(36 - 16 = 20\).
• Simplify the denominator: \(4 - 6 = -2\).

Then, we divide 20 by -2 to get -10.
Key points to remember when dividing rational numbers include:

• Always simplify the numerator and denominator first.
• Be attentive to signs (positive or negative). Dividing a positive number by a negative number results in a negative quotient.

Practicing these steps makes dividing rational numbers a straightforward process.