Problem 293 - Elementary Algebra

Question

In the following exercises, simplify. $$ (10-\sqrt{3})(10+\sqrt{3}) $$

Step-by-Step Solution

Answer

97

Step 1: Identify the Expression

The given expression is \((10-\sqrt{3})(10+\sqrt{3})\). This is a difference of squares situation.

Step 2: Apply the Difference of Squares Formula

Recall that \(a^{2} - b^{2} = (a-b)(a+b)\), where \(a = 10\) and \(b = \sqrt{3}\). Thus, \((10-\sqrt{3})(10+\sqrt{3}) = 10^{2} - (\sqrt{3})^{2}\).

Step 3: Calculate the Squares

Calculate \(10^{2} = 100\) and \( (\sqrt{3})^{2} = 3 \).

Step 4: Simplify the Expression

Now, substitute back into the difference of squares formula: \(100 - 3 = 97\).

Key Concepts

Difference of squares

The expression \( (10-\sqrt{3})(10+\sqrt{3}) \) is a classic example of the difference of squares. This occurs when you have the product of a binomial and its conjugate. In general, the formula for the difference of squares is \( a^{2} - b^{2} = (a-b)(a+b) \). We use this formula because the middle terms cancel out, making it easier to simplify the expression.
In this exercise, \( a = 10 \) and \( b = \sqrt{3} \). Applying the formula, we get:
\[ (10 - \sqrt{3})(10 + \sqrt{3}) = 10^{2} - (\sqrt{3})^{2} \].
By recognizing patterns like these, you can quickly simplify complex expressions.

Algebraic formulas

Algebraic formulas help simplify and solve problems quickly. One commonly used formula is the difference of squares, as discussed above. In addition to this, you should familiarize yourself with other key formulas:

• Square of a sum: \( (a+b)^{2} = a^{2} + 2ab + b^{2} \)
• Square of a difference: \( (a-b)^{2} = a^{2} - 2ab + b^{2} \)
• Product of a sum and difference: \( (a+b)(a-b) = a^{2} - b^{2} \)

These are derived from multiplying out binomials and recognizing patterns in the terms. Practicing these formulas makes working with polynomials easier.
In our specific exercise, identifying the difference of squares formula allowed us to simplify \( 10^{2} - (\sqrt{3})^{2} \) directly.

Radicals in algebra

Radicals often appear in algebra, and understanding how to simplify them is crucial. A radical generally represents the root of a number. For instance, \( \sqrt{3} \) is the square root of 3.
Here are some important points about radicals:

• \( \sqrt{a^{2}} = a \)
• \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)
• Rationalizing the denominator involves removing the radical from the denominator by multiplying the numerator and denominator by a suitable value.

In our exercise, we dealt with \( \sqrt{3} \). We calculated \( (\sqrt{3})^{2} = 3 \), which is essential in the simplification process. By recognizing how to handle radicals, you can solve many problems in algebra more efficiently.