Problem 33 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (\sqrt{3}-\sqrt{7})(\sqrt{3}+2 \sqrt{7}) $$

Step-by-Step Solution

Answer

The simplified expression is -4 or -25, depending on whether b equals \(\sqrt{7}\) or \(2\sqrt{7}\)

Step 1: Identify the difference of squares formula

Observe the expression \((\sqrt{3}-\sqrt{7})(\sqrt{3}+2 \sqrt{7})\). This is in the form of \((a - b)(a + b)\), which is the difference of squares formula. The formula says that \((a^2 - b^2)\) is equal to \((a - b)(a + b)\). In this case, \(a = \sqrt{3}\) and \(b = -\sqrt{7}\) or \(2\sqrt{7}\).

Step 2: Apply the difference of squares formula

Applying the difference of squares formula, \((a - b)(a + b)\) simplifies to \((a^2 - b^2)\). Therefore, \((\sqrt{3}-\sqrt{7})(\sqrt{3}+2 \sqrt{7})\) simplifies to \((\sqrt{3})^2 - (\sqrt{7})^2\) or \((\sqrt{3})^2 - (2\sqrt{7})^2\).

Step 3: Compute the squares and subtract

Compute the squares. Remember that \((\sqrt{n})^2 = n\). \((\sqrt{3})^2\) equals 3 and \((\sqrt{7})^2\) equals 7 and \((2\sqrt{7})^2\) equals 28. Then subtract the two results to get the simplified expression.

Step 4: Write the final, simplified expression

After doing the subtraction, the final, simplified expression is -4 or -25 depending on whether \(b = \sqrt{7}\) or \(2\sqrt{7}\).

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify expressions quickly. It is based on the formula \((a - b)(a + b) = a^2 - b^2\). You can recognize this pattern in any expression where you have two binomials, one subtracted from and the other added to the same central term.
For example, consider an expression such as \((abla - riangle)(abla + riangle)\). Applying the difference of squares, this expression becomes \(abla^2 - riangle^2\).
In simpler terms:

• Identify parts of the expression that can be paired up as \(a - b\) and \(a + b\).
• Solve for \(a^2\) and \(b^2\) separately.
• Subtract \(b^2\) from \(a^2\) to simplify the expression.

By recognizing and applying this structure, you can simplify many algebraic expressions with ease.

Rationalization

Rationalization involves eliminating radicals from the denominator of an expression. When faced with a fraction containing a square root or other irrational numbers, it's essential to convert them into a rational form for easy interpretation.
The typical method is to multiply both the numerator and the denominator by a conjugate or a suitable radical expression. This approach helps remove the radical from the denominator.
For example, if you have an expression such as \(\frac{1}{\sqrt{a}}\), rationalize by multiplying by \(\frac{\sqrt{a}}{\sqrt{a}}\), resulting in \(\frac{\sqrt{a}}{a}\).
Key steps include:

• Identify the radical in the denominator.
• Multiply by the conjugate or a matching radical over itself (i.e., form \(\frac{\text{conjugate}}{\text{conjugate}}\)).
• Simplify both the top and bottom of the fraction.

This process ensures that the denominator becomes a rational number, making it easier to understand and work with.

Simplifying Expressions

Simplifying expressions is about reducing them into their simplest form, making them easier to understand and work with. This involves several strategies, such as combining like terms, applying arithmetic operations, and using algebraic identities.
To simplify a radical expression, you will often employ methods like factorization, distribution, or the application of identities, such as the difference of squares.
Here’s a simple guideline for simplifying expressions:

• Identify and combine like terms where possible.
• Use algebraic identities, like \((a-b)(a+b)=a^2-b^2\), to simplify.
• Reduce fractions and evaluate any possible arithmetic operations.

By simplifying expressions, you not only make them more manageable but also open the door to deeper comprehension of the underlying mathematical concepts.