Problem 273 - Elementary Algebra

Question

In the following exercises, simplify. $$ (7+\sqrt{3})(9-\sqrt{3}) $$

Step-by-Step Solution

Answer

The simplified form is \( 60 + 2\sqrt{3} \).

Step 1: Apply the Distributive Property

To simplify \( (7+\sqrt{3})(9-\sqrt{3}) \), use the distributive property (also known as the FOIL method for binomials) to expand the expression. This means multiplying each term in the first binomial by each term in the second binomial: \(7*9 + 7*(-\sqrt{3}) + \sqrt{3}*9 + \sqrt{3}*(-\sqrt{3})\).

Step 2: Perform the Multiplications

Carry out each of the multiplications from Step 1: \( 7*9 = 63 \), \( 7*(-\sqrt{3}) = -7\sqrt{3} \), \( \sqrt{3}*9 = 9\sqrt{3} \), and \( \sqrt{3}*(-\sqrt{3}) = -3 \).

Step 3: Combine Like Terms

Add all the terms from Step 2 together: \( 63 - 7\sqrt{3} + 9\sqrt{3} - 3 \). Combine the like terms \( 63-3 \) and \( -7\sqrt{3} + 9\sqrt{3} \). This simplifies to \( 60 + 2\sqrt{3} \).

Key Concepts

distributive property

The Distributive Property is a fundamental concept in algebra. It states that for any numbers a, b, and c, the expression \(a(b + c)\) is equivalent to \(ab + ac\). When applied to binomials, this method is often called the FOIL method, which stands for First, Outside, Inside, Last. Here, we multiply each term in the first binomial by each term in the second binomial.

To simplify the expression \((7+\sqrt{3})(9-\sqrt{3})\), we distribute each term in \(7+\sqrt{3}\) to each term in \(9-\sqrt{3}\). This gives us: - Multiply the first terms: \(7 * 9 = 63\) - Multiply the outside terms: \(7 * (-\sqrt{3}) = -7\sqrt{3}\) - Multiply the inside terms: \(\root{3} * 9 = 9\root{3}\) - Multiply the last terms: \(\root{3} * (-\root{3}) = -3\)

Now, let's move on to simplifying these multiplied expressions in the next section.

simplifying expressions

Simplifying algebraic expressions involves performing all multiplications and combining all like terms to create a more straightforward expression. From our distribution, we have: \(63, -7\root{3}, 9\root{3},\) and \(-3\). Now, handle each multiplication separately:

• First, calculate the integers: \(7 * 9 = 63\)
• Next, distribute the numbers involving square roots: \(7 * (-\root{3}) = -7\root{3}\) and \(\root{3} * 9 = 9\root{3}\)
• Finally, simplify the product of the square roots: \(\root{3} * (-\root{3}) = -3\).

Now we have the expanded terms: \(63, -7\root{3}, 9\root{3}, \) and \(-3\). Remember, to simplify expressions, always carefully perform the multiplications step-by-step before moving on.

combining like terms

Combining Like Terms is the process where we simplify algebraic expressions by adding or subtracting terms that have the same variables raised to the same power. After distribution and simplification, we have these terms from our example: \(63, -7\root{3}, 9\root{3}, \) and \(-3\).

Combine the constant terms first: \(63 - 3 = 60\).

Next, combine the terms with the square root: \(-7\root{3} + 9\root{3} = 2\root{3}\). If you look closely, these operations involve basic addition and subtraction. When terms are 'like', it means they contain the same variable or root, and hence can be combined directly.

So, after combining like terms and simplifying the expressions, our final result is: \(60 + 2\root{3}\).

This is a simpler, more concise version of our original expression.