Simplify $(3+\sqrt{7})(3-\sqrt{7})$: A Math Problem

When we first encounter a mathematical expression like $(3+\sqrt{7})(3-\sqrt{7})$, it might look a little intimidating, especially if you're not a big fan of square roots. But don't worry, this is actually a pretty straightforward problem that uses a common algebraic identity. The core idea here is to find the product of $(3+\sqrt{7})(3-\sqrt{7})$. This means we need to multiply these two binomials together. There are a few ways to approach this, but the most efficient method relies on recognizing a pattern. Let's break down why this expression is special and how to solve it step-by-step.

Understanding the Algebraic Identity

The expression $(3+\sqrt{7})(3-\sqrt{7})$ is a classic example of the difference of squares formula. This formula states that for any two numbers, a and b, the product of their sum and their difference is equal to the square of the first number minus the square of the second number. In mathematical terms, this is written as: $(a+b)(a-b) = a^2 - b^2$. In our specific problem, a is 3 and b is $\sqrt{7}$. Recognizing this pattern is key to simplifying the expression quickly and avoiding unnecessary calculations. If we didn't spot this, we could use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) to multiply the binomials, but that would involve more steps and a higher chance of making a mistake.

Step-by-Step Solution Using the Difference of Squares

Let's apply the difference of squares formula to our problem: $(3+\sqrt{7})(3-\sqrt{7})$.

1. Identify a and b: In this case, $a = 3$ and $b = \sqrt{7}$.
2. Apply the formula $a^2 - b^2$: Substitute our values into the formula.
   • $a^2 = 3^2 = 9$
   • $b^2 = (\sqrt{7})^2 = 7$
3. Calculate the result: $a^2 - b^2 = 9 - 7 = 2$.

So, the product of $(3+\sqrt{7})(3-\sqrt{7})$ is simply 2. This is a much simpler answer than the options provided, which suggests we should carefully examine how those options are derived, perhaps through incorrect application of the distributive property or misunderstanding the nature of square roots.

Examining the Given Options

Let's look at the options provided to see how they relate to the problem, and why they are incorrect:

• Option A: $6-3 \sqrt{7}+3 \sqrt{7}-\sqrt{14}$ This option seems to have resulted from a misunderstanding of multiplication. If we were to expand $(3+\sqrt{7})(3-\sqrt{7})$ using FOIL, the steps would be:

  • First: $3 \times 3 = 9$
  • Outer: $3 \times -\sqrt{7} = -3\sqrt{7}$
  • Inner: $\sqrt{7} \times 3 = 3\sqrt{7}$
  • Last: $\sqrt{7} \times -\sqrt{7} = -(\sqrt{7})^2 = -7$ Adding these together gives $9 - 3\sqrt{7} + 3\sqrt{7} - 7$. Notice how the middle terms $(-3\sqrt{7} + 3\sqrt{7})$ cancel out to zero, leaving $9 - 7 = 2$. Option A incorrectly has $-\sqrt{14}$ as the last term, which doesn't follow from multiplying $\sqrt{7}$ by itself. It might stem from incorrectly multiplying $\sqrt{7}$ by $\sqrt{7}$ as $\sqrt{14}$ or some other faulty logic.

• Option B: $9-\sqrt{21}+\sqrt{21}-\sqrt{49}$ This option also shows signs of incorrect multiplication. The first term, 9, is correct ($3 \times 3$). The second term, $-\sqrt{21}$, might come from multiplying 3 by $-\sqrt{7}$, which should be $-3\sqrt{7}$. The third term, $+\sqrt{21}$, might come from multiplying $\sqrt{7}$ by 3, which should be $+3\sqrt{7}$. The combination of $\sqrt{21}$ is incorrect, as it should be $3\sqrt{7}$. Furthermore, the last term, $-\sqrt{49}$, is equal to $-7$, which is the correct value for the last term in the FOIL expansion, but the middle terms are incorrectly represented as $\sqrt{21}$ instead of $3\sqrt{7}$. The problem here lies in how the terms involving square roots were handled.

• Option C: $9-3 \sqrt{7}+3 \sqrt{7}-\sqrt{49}$ This option is the closest to the correct FOIL expansion. Let's break it down:

  • First: $3 \times 3 = 9$. (Correct)
  • Outer: $3 \times -\sqrt{7} = -3\sqrt{7}$. (Correct)
  • Inner: $\sqrt{7} \times 3 = 3\sqrt{7}$. (Correct)
  • Last: $\sqrt{7} \times -\sqrt{7} = -(\sqrt{7})^2 = -7$. This is where the option deviates. It writes $-\sqrt{49}$. While $\sqrt{49}$ is indeed 7, and therefore $-\sqrt{49}$ is $-7$, the step of writing it as $-\sqrt{49}$ instead of directly as $-7$ is slightly unusual in this context, though mathematically equivalent. If we were to simplify this option completely, we would get $9 - 3\sqrt{7} + 3\sqrt{7} - 7$. The middle terms $-3\sqrt{7} + 3\sqrt{7}$ cancel out to 0, leaving $9 - 7 = 2$. So, even though the representation of the last term is slightly different, the result of simplifying Option C would be 2, matching our difference of squares calculation.

Conclusion

The problem asks us to choose the product of $(3+\sqrt{7})(3-\sqrt{7})$. As we've shown using the difference of squares formula, the product is $3^2 - (\sqrt{7})^2 = 9 - 7 = 2$. None of the options provided directly equal 2 in their initial form, but Option C, $9-3 \sqrt{7}+3 \sqrt{7}-\sqrt{49}$, simplifies to 2. Therefore, if the question is asking for the expanded form before final simplification, Option C is the correct representation of the FOIL expansion, with the last term written as $-\sqrt{49}$ instead of $-7$. It's crucial to understand that the goal is to find the simplest form of the product, which is 2. However, based on the structure of the options, it seems the question is looking for the intermediate step of expansion.

For further exploration of algebraic identities and simplifying radicals, you can visit resources like Khan Academy. They offer excellent explanations and practice problems on these topics.

• Khan Academy - Algebra