Problem 274 - Elementary Algebra

Question

In the following exercises, simplify. $$ (8-\sqrt{2})(3+\sqrt{2}) $$

Step-by-Step Solution

Answer

22 + 5 \sqrt{2}

Step 1 - Apply the Distributive Property

Use the distributive property \(a(b+c) = ab + ac\) to multiply each term in the first binomial by each term in the second binomial. This gives: \[(8 - \sqrt{2})(3 + \sqrt{2}) = 8 \cdot 3 + 8 \cdot \sqrt{2} - \sqrt{2} \cdot 3 - \sqrt{2} \cdot \sqrt{2}\]

Step 2 - Multiply the Terms

Multiply each pair of terms obtained from the distributive property: \[8 \cdot 3 + 8 \cdot \sqrt{2} - \sqrt{2} \cdot 3 - \sqrt{2} \cdot \sqrt{2}\] becomes: \[24 + 8 \sqrt{2} - 3 \sqrt{2} - 2\]

Step 3 - Combine Like Terms

Combine the constant terms and the terms with \(\sqrt{2}\): \[24 - 2 + 8 \sqrt{2} - 3 \sqrt{2}\] which simplifies to \[22 + 5 \sqrt{2}\]

Key Concepts

distributive property

To start simplifying the expression \(8-\sqrt{2})(3+\sqrt{2})\), we use the distributive property. The distributive property allows us to multiply each term in the first binomial by each term in the second binomial. This is expressed as \(a(b+c) = ab + ac\). In our case, we distribute each term in \(8-\sqrt{2}\) to each term in \(3+\sqrt{2}\). This results in: \[ (8 - \sqrt{2})(3 + \sqrt{2}) = 8 \cdot 3 + 8 \cdot \sqrt{2} - \sqrt{2} \cdot 3 - \sqrt{2} \cdot \sqrt{2} \] Breaking this down:

• Multiply \(8\) by \(3\)
• Multiply \(8\sqrt{2}\)
• Multiply \(-\sqrt{2}\) with \(3\)
• Multiply \(-\sqrt{2}\) with \(\sqrt{2}\)

This step helps in breaking the expression into simpler parts.

combining like terms

After applying the distributive property and multiplying the terms, we get: \[ 24 + 8 \sqrt{2} - 3 \sqrt{2} - 2 \] Now we can combine the like terms. In algebra, like terms are terms that have the same variables raised to the same power. For our expression:

• The constant terms are \24\ and \-2\.
• The like terms with \sqrt{2}\ are \8\sqrt{2}\ and \-3\sqrt{2}\.

Combining the constant terms and the like terms gives: \[ 24 - 2 + 8 \sqrt{2} - 3 \sqrt{2} = 22 + 5 \sqrt{2} \] This is the simplified form of the expression.

binomial multiplication

To simplify an expression like \(8-\sqrt{2})(3+\sqrt{2})\), we engage in binomial multiplication. When multiplying binomials, we apply the distributive property, multiplying every term in the first binomial by every term in the second binomial. This process can also be noted as the FOIL method, where we multiply:

• First terms
• Outer terms
• Inner terms
• Last terms

This is a systematic way to ensure that no terms are missed. For our exercise, after using the distributive method: \[8 \cdot 3 + 8 \cdot \sqrt{2} - \sqrt{2} \cdot 3 - \sqrt{2} \cdot \sqrt{2}\] we arrive at: \[ 24 + 8 \sqrt{2} - 3 \sqrt{2} - 2 \] Understanding binomial multiplication simplifies complex expressions into manageable components, making it easier to combine like terms and arrive at the simplified form.