Problem 75 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \frac{2}{3-\sqrt{7}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression $$\frac{2}{3-\sqrt{7}}$$ is $$3+\sqrt{7}$$.

Step 1: Identify the conjugate

First, we will identify the conjugate of the given binomial expression. The conjugate of $$3-\sqrt{7}$$ is $$3+\sqrt{7}$$.

Step 2: Multiply by conjugate

Next, we will multiply both the numerator and denominator of the given fraction by the conjugate we found in the previous step. $$ \frac{2}{3-\sqrt{7}} \cdot \frac{3+\sqrt{7}}{3+\sqrt{7}} $$

Step 3: Apply FOIL method

Now, we will use the FOIL (First, Outer, Inner, Last) method to multiply the terms in the numerator and denominator of the resulting fraction. $$ \frac{2(3+\sqrt{7})}{(3-\sqrt{7})(3+\sqrt{7})} $$

Step 4: Simplify numerator and denominator

Next, we will simplify the numerator and denominator of the resulting fraction as follows: Numerator: $$ 2(3+\sqrt{7}) = 6+2\sqrt{7} $$ Denominator: $$ (3-\sqrt{7})(3+\sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9-7 $$

Step 5: Write final simplified expression

Finally, we substitute the simplified numerator and denominator, and simplify the resulting fraction: $$ \frac{6+2\sqrt{7}}{9-7} = \frac{6+2\sqrt{7}}{2} $$ Since the numerator and denominator have a common factor of 2, we can simplify further: $$ \frac{6+2\sqrt{7}}{2}=\frac{2(3+\sqrt{7})}{2} = 3+\sqrt{7} $$ The final simplified expression is $$3+\sqrt{7}$$.

Key Concepts

Conjugates

In mathematics, the conjugate of a binomial expression is a specific way to manipulate radicals that appear in fractions. A conjugate involves changing the sign between two terms. For instance, if you have a binomial expression like \( a - b \), its conjugate would be \( a + b \). The special property of conjugates is that when multiplied by each other, the middle term cancels out due to differences of squares. This technique is crucial in rationalizing denominators when you want to eliminate radicals, making expressions simpler to work with.

• The given binomial: \( 3 - \sqrt{7} \)
• Its conjugate: \( 3 + \sqrt{7} \)

By multiplying a fraction by a conjugate, you're effectively clearing the radical from the denominator without changing the value of the original expression.

FOIL Method

The FOIL Method, which stands for First, Outer, Inner, Last, is a popular technique for multiplying two binomials. It's a systematic approach that ensures all components are correctly accounted for during multiplication, which can be especially helpful for students to visualize all parts of the expression.

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

In the given exercise, using the FOIL method on the denominator \((3 - \sqrt{7})(3 + \sqrt{7})\) helps us simplify this expression to a difference of squares: \(3^2 - (\sqrt{7})^2\), resulting in \(9 - 7\). The FOIL method is especially useful in expanding expressions to observe all terms involved.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form where all possible arithmetic operations have been performed. This process often involves combining like terms, factoring, or employing techniques like rationalizing.

• After multiplying by the conjugate, we simplify the numerator to \(6 + 2\sqrt{7}\).
• The denominator is simplified using differences of squares, becoming \(2\).

Next, simplify the fraction \(\frac{6 + 2\sqrt{7}}{2}\) by canceling out the common factor, which in this case is \(2\). Simplifying expressions makes them easier to handle, interpret, and verify, while also ensuring they are presented in the clearest manner possible.

Binomials

Binomials are simple mathematical expressions with two terms, usually joined by a plus or a minus sign. Understanding the structure of binomials is crucial for operations like multiplication or finding conjugates.

• Consider the binomial: \(3 - \sqrt{7}\)
• Its conjugate: \(3 + \sqrt{7}\)

These expressions are foundational elements in algebra, often appearing in scenarios like polynomial multiplication or rationalizing denominators. The operation involving the conjugate is a direct application of learning how to manage binomials. Recognizing a binomial and its characteristics lets you predict the outcome of various operations and simplifies manipulating expressions during algebraic operations.