Problem 36 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (2-\sqrt{98})(3+\sqrt{18}) $$

Step-by-Step Solution

Answer

The simplified form of the expression \((2-\sqrt{98})(3+\sqrt{18})\) is \(6 - 57\sqrt{2}\).

Step 1: Expand the expression

Start by expanding the expression using the distributive property, also known as the FOIL method (First, Outer, Inner, Last): \((a+b)(c+d) = ac + ad + bc + bd\). This gives: \((2-\sqrt{98})(3+\sqrt{18}) = 6 + 2\sqrt{18} - 3\sqrt{98} - \sqrt{98 \cdot 18}\).

Step 2: Simplify the square roots

Now, simplify each of the square roots. We have \(\sqrt{18} = \sqrt{2 \cdot 9} = 3\sqrt{2}\), \(\sqrt{98} = \sqrt{2 \cdot 49} = 7\sqrt{2}\), and \(\sqrt{98 \cdot 18} = \sqrt{2 \cdot 2 \cdot 49 \cdot 9} = 21\sqrt{8}\). Substitute these values back into the expanded expression from Step 1, leading to: \(6 + 2 \cdot 3\sqrt{2} - 3 \cdot 7\sqrt{2} - 21\sqrt{8}\).

Step 3: Combine like terms

Now, combine like terms. This gives us \(6 + 6\sqrt{2} - 21\sqrt{2} - 21\sqrt{8}\). This can be further simplified by combining the terms involving the square root of 2, and by simplifying the square root of 8 = \sqrt{2 \cdot 4} = 2\sqrt{2}, leading to: \(6 -15\sqrt{2} - 42\sqrt{2}\).

Step 4: Final Simplification

Combining final like terms, we get: \(6 - 57\sqrt{2}\). This is the fully simplified expression.

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra, allowing us to simplify expressions where one term is multiplied by a sum or difference of terms. In our exercise, we applied the distributive property to expand

• (2 - \(\sqrt{98}\))(3 + \(\sqrt{18}\)) into separate terms

The distributive property can be thought of as spreading out terms:

• Multiply the first term of one binomial by each term of the other binomial.
• Then, do the same for the second term.

The structure can be remembered using the acronym FOIL:

• First,
• Outer,
• Inner,
• Last.

For instance, in this problem we applied it as follows: \[(2 \cdot 3) + (2 \cdot \sqrt{18}) - (\sqrt{98} \cdot 3) - (\sqrt{98} \cdot \sqrt{18})\]. This gives us our expanded form of the equation, enabling us to simplify further.

Simplifying Square Roots

Simplifying square roots involves finding the prime factors and pulling out any pairs as single numbers outside the square root. It's an effective method for simplifying expressions and making calculations manageable. In our exercise, we simplify roots like \(\sqrt{18}\), \(\sqrt{98}\), and \(\sqrt{98 \cdot 18}\). Here's how:

• For \(\sqrt{18}\): Recognize this as \(\sqrt{2 \cdot 9}\). Pull the square of 3 out, which gives you \(3\sqrt{2}\).
• For \(\sqrt{98}\): Break it down into \(\sqrt{2 \cdot 49}\). Since 49 is a square of 7, it simplifies to \(7\sqrt{2}\).
• For \(\sqrt{98 \cdot 18}\): Recognize that simplifying directly might be complex; first simplify the separate square roots before multiplication.

This transformation of the square roots was key to progressing through the problem.

Combining Like Terms

Combining like terms is crucial in simplifying algebraic expressions. It involves summing up coefficients of terms that have the same variable parts. In this exercise, we combined terms involving square roots.

• The terms \(6\sqrt{2}\) and \(-21\sqrt{2}\) were combined to form \(-15\sqrt{2}\).
• By simplifying \(\sqrt{8} = 2\sqrt{2}\), we further combined it with \(-21\) resulting in \(-42\sqrt{2}\).

This process helps in reducing the equation to its neatest form, making \\(6 - 57\sqrt{2}\)\, our simplified expression. Consistently, combining like terms ensures correctness and precision in algebraic solutions.