Problem 309 - Elementary Algebra

Question

In the following exercises, simplify. $$ (-4+\sqrt{17})(-3+\sqrt{17}) $$

Step-by-Step Solution

Answer

29 - 7\text{√}{17}

Step 1: Apply the Distributive Property

Use the distributive property \((a+b)(c+d) = ac + ad + bc + bd\) to multiply \((-4+\text{√}{17})(-3+\text{√}{17})\).

Step 2: Multiply Each Term

Multiply each term in the first binomial by each term in the second binomial: \((-4 \times -3) + (-4 \times \text{√}{17}) + (\text{√}{17} \times -3) + (\text{√}{17} \times \text{√}{17})\).

Step 3: Simplify the Products

Calculate the individual products: \(-4 \times -3 = 12\), \(-4 \times \text{√}{17} = -4\text{√}{17}\), \(\text{√}{17} \times -3 = -3\text{√}{17}\), and \(\text{√}{17} \times \text{√}{17} = 17\).

Step 4: Combine Like Terms

Combine the calculated products: \(12 - 4\text{√}{17} - 3\text{√}{17} + 17\)

Step 5: Final Simplification

Add and combine like terms: \(12 + 17 - 4\text{√}{17} - 3\text{√}{17} = 29 - 7\text{√}{17}\)

Key Concepts

distributive property

To understand algebra simplification, the distributive property is a good starting point. The distributive property states that \( a(b+c) = ab + ac \). This means you distribute the term outside the parenthesis to each term inside the parenthesis.

In our exercise, we have: \((-4 + \sqrt{17})(-3 + \sqrt{17})\).
By using the distributive property, we distribute each term in the first binomial, \((-4 + \sqrt{17})\), to each term in the second binomial, \((-3 + \sqrt{17})\). This breaks down the multiplication into simpler steps, which we can solve individually.

binomial multiplication

Binomial multiplication is an extension of the distributive property, applied when we have two binomials.

In the exercise, we multiplied each term in the first binomial by each term in the second binomial.
For example, \( (-4 + \sqrt{17})(-3 + \sqrt{17})\) results in 4 separate products:

• \(-4 \times -3\)
• \(-4 \times \sqrt{17}\)
• \(\sqrt{17} \times -3\)
• \(\sqrt{17} \times \sqrt{17}\)

This ensures all combinations are accounted for before we proceed to combine them.

combining like terms

After multiplying the binomials, our next step is to combine like terms. Like terms are terms that have the same variable raised to the same power

From the multiplication, we have: \(12, -4\sqrt{17}, -3\sqrt{17},\) and \(17\).

• Combine the constants \(12 + 17\), which equals \(29\).
• Combine the square root terms: \(-4\sqrt{17} - 3\sqrt{17}\), which equals \(-7\sqrt{17}\).

Our simplified expression is \( 29 - 7\sqrt{17} \). Combining like terms helps to simplify the expression into its most reduced form.

square roots in algebra

Dealing with square roots in algebra often requires understanding their properties. For instance, \(\sqrt{a} \times \sqrt{a} = a\). In our case, \(\sqrt{17} \times \sqrt{17} = 17\).

Working with square roots also means recognizing them as like terms when they have the same radicand.
In our exercise, \(-4\sqrt{17} \) and \(-3\sqrt{17}\) are like terms that can be combined easily.