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Chapter 7: Problem 54

Simplify the products. Give exact answers. $$(\sqrt{5}+2)(\sqrt{5}-6)$$

Short Answer

Expert verified
Answer Step Summarized 13shorterthan

Step by step solution

01

Expand the Binomials

Use the distributive property (FOIL method) to expand the expression ( ((\begin{array}{l}((\begin{array}{ll}\begin{array}{l}\begin{array}{l}(\begin{array}{ll}\begin{array}{l}(\sqrt{5}{5}+2)(\sqrt{5}5\-6\begin{array}{ll}\begin{array}{l}}}Eeq}{30} will not be using dollars for equations instead we will use \(...\) or \[...\].sheep expansion method obtained position 2 method achievinga_n}} \times smalleruars two linear functions five calculated
02

Combine Like Terms

After expanding all the terms, combine the like terms. combining expression - 2derivitives of different types yield together to form similarform exactcoordinates
03

- Perform Final Simplification

In the final step, simplify each of the like terms. term to a final simplifiedformat. theidentical termsare grouped

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
To begin simplifying the expression \( (\sqrt{5} + 2)(\sqrt{5} - 6) \), we use the distributive property. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In our case, we apply this method to expand our product, better known as the FOIL method for binomials (First, Outer, Inner, Last) method. First, we multiply \( \sqrt{5} \times \sqrt{5} \), then the outer terms \( \sqrt{5} \times (-6) \), followed by the inner terms \( 2 \times \sqrt{5} \), and finally the last terms \( 2 \times (-6) \). This method breaks down a complex expression into simpler parts that can be more easily managed.
Combining Like Terms
After expanding the binomial, the expression now has several terms: \( 5 - 6\sqrt{5} + 2\sqrt{5} - 12 \). The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our case, we identify \( -6\sqrt{5} \) and \( 2\sqrt{5} \) as like terms. By adding them together, we get their combined value: \( -6\sqrt{5} + 2\sqrt{5} = -4\sqrt{5} \). This makes the expression simpler and shorter.
Binomial Expansion
Binomial expansion is the method we used to turn \( (\sqrt{5} + 2)(\sqrt{5} - 6) \) into the expanded form \( 5 - 6\sqrt{5} + 2\sqrt{5} - 12 \). Each term in the binomial contributes to the expanded format when multiplied out. This process simplifies handling complex expressions. When properly carried out, it provides a structured approach to dealing with polynomial multiplication, especially helpful for students to grasp as test questions frequently use binomials in algebra.
Final Simplification
In the final simplification stage, ensure all like terms are combined and constants are appropriately summed. From our combined like terms, the expression \( 5 - 4\sqrt{5} - 12 \) turns into \( -7 - 4\sqrt{5} \) after summing up \( 5 - 12 \). This is the simplest version of the given complex expression, making it easier to understand and utilize in further equations or transformations.

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Most popular questions from this chapter

Simplify. $$\frac{5+\sqrt{75}}{10}$$

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According to the new International America's Cup Class Rules, the maximum sail area in square meters for a yacht in the America's Cup race is given by the function $$S=\left(13.0368+7.84 D^{1 / 3}-0.8 L\right)^{2}$$ where \(D\) is the displacement in cubic meters \(\left(\mathrm{m}^{3}\right),\) and \(L\) is the length in meters (m). (www.sailing.com). Find the maximum sail area for a boat that has a displacement of \(18.42 \mathrm{m}^{3}\) and a length of \(21.45 \mathrm{m}\).

Solve each equation. $$(x-3)^{2 / 3}=-4$$
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