Problem 276 - Elementary Algebra

Question

In the following exercises, simplify. $$ (3-\sqrt{7})(5-\sqrt{7}) $$

Step-by-Step Solution

Answer

Short answer here

Step 1: Use the distributive property

To multiply the two binomials, apply the distributive property:

Step 2: Multiply each term

Multiply each term in the first binomial by each term in the second binomial:

Step 3: Simplify the expression

Combine like terms to simplify the expression:

Key Concepts

Distributive Property

The distributive property is a key concept in algebra. It's useful when you need to multiply one term by each term in a parenthesis. When you see an expression like \(a(b + c)\), the distributive property lets you rewrite it as \(ab + ac\). This technique helps simplify complex expressions.
For example, when multiplying two binomials, \((3-\sqrt{7})(5-\sqrt{7})\), you first apply the distributive property: you multiply each term in the first binomial by each term in the second binomial.

Multiplying Binomials

Multiplying binomials requires using the distributive property twice. It involves four multiplications in total. For the given problem, you multiply each term in the first binomial \((3-\sqrt{7})\) by each term in the second binomial \((5-\sqrt{7})\).
Here's how:
\[ = 3 \cdot 5 + 3 \cdot (-\sqrt{7}) + (-\sqrt{7}) \cdot 5 + (-\sqrt{7}) \cdot (-\sqrt{7}) \]

Breaking it down step-by-step makes it manageable:

• First multiply 3 and 5 to get 15.
• Then multiply 3 and \(-\sqrt{7}\) to get \-3\sqrt{7}\.
• After that, multiply \(-\sqrt{7}\) and 5 to get \-5\sqrt{7}\.

This exercise uses multiplication rules straightforwardly, contributing to simpler expressions.

Combining Like Terms

After multiplying out the binomials, the next step is to combine like terms. Like terms have the same variables raised to the same power. In this case, combining like terms simplifies your expression.

From the previous section, we had \[15 - 3\sqrt{7} - 5\sqrt{7} +(\sqrt{7})^2 \]

• First, notice that \(\sqrt{7} \cdot \sqrt{7} \) simplifies to \ 7\.
• Adding these gives \[15 - 3\sqrt{7} - 5\sqrt{7} + 7 \].
• Then, combine the constant terms 15 and 7 to get 22.
• Lastly, simplify \(- 3 + (- 5) = - 8\sqrt{7} \).

This makes your final simplified expression \[ = 22 - 8\sqrt{7} \]. Taking many steps makes the expression clearer and simpler.