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Chapter 10: Problem 62

Multiply, and then simplify if possible. \((\sqrt{6}-4 \sqrt{2})(3 \sqrt{6}+\sqrt{2})\)

Short Answer

Expert verified
The simplified expression is \(10 - 22\sqrt{3}\).

Step by step solution

01

Distribute Terms

We begin by using the distributive property. The expression is \[(\sqrt{6} - 4\sqrt{2})(3\sqrt{6} + \sqrt{2}),\]which should be expanded by multiplying each term in the first binomial by each term in the second binomial:\[\sqrt{6}(3\sqrt{6}) + \sqrt{6}(\sqrt{2}) - 4\sqrt{2}(3\sqrt{6}) - 4\sqrt{2}(\sqrt{2}).\]
02

Simplify Terms

Simplify each term from the expansion:- \(\sqrt{6} \cdot 3 \sqrt{6} = 3 \cdot 6 = 18\).- \(\sqrt{6} \cdot \sqrt{2} = \sqrt{12} = 2 \sqrt{3}\), since \(\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}\).- \(-4\sqrt{2} \cdot 3\sqrt{6} = -12 \cdot \sqrt{12} = -24 \sqrt{3}\), where \(\sqrt{12} = 2 \sqrt{3}\).- \(-4\sqrt{2} \cdot \sqrt{2} = -4 \cdot \sqrt{4} = -4 \cdot 2 = -8\).
03

Combine Like Terms

Combine all the simplified terms obtained from Step 2:- From \(18 + 2\sqrt{3} - 24\sqrt{3} - 8\).- The constants combine as \(18 - 8 = 10\).- Combine the square root terms: \(2\sqrt{3} - 24\sqrt{3} = -22\sqrt{3}\).- Thus, the expression simplifies to \(10 - 22\sqrt{3}\).

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Key Concepts

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

Distributive Property
In algebra, the distributive property is an important tool for expanding and simplifying expressions. It allows us to distribute one value across terms inside a parenthesis. The distributive property can be written as:
  • When distributing over addition: \(a(b + c) = ab + ac\)
  • When distributing over subtraction:\(a(b - c) = ab - ac\)
In this exercise, the distributive property is applied to \((\sqrt{6} - 4\sqrt{2})(3\sqrt{6} + \sqrt{2})\).We multiply each term in the first binomial by each term in the second binomial, leading to four separate products. The careful application of the distributive property sets the groundwork for solving the problem effectively, allowing us to handle complex expressions involving radicals.
Simplifying Radicals
Simplifying radicals helps reduce expressions to their simplest form, making them easier to work with. When we have a radical, it鈥檚 often possible to break it down or 'simplify' it.
  • Identify any perfect square factors in the radicand (the number inside the radical).
  • Express the radical as the product of two square roots: one involving perfect squares, if possible.
  • Simplify the square root of the perfect square factor.
For instance, in our exercise \(\sqrt{12}\) is simplified to \(2\sqrt{3}\), as \(\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4}\sqrt{3} = 2\sqrt{3}\).This simplification process is key to combining terms effectively later, especially when dealing with similar radical terms.
Binomial Multiplication
Binomial multiplication involves the multiplication of two expressions that each contain two terms. This typically results in four products, which can be simplified and combined. In this case, the original expression was \((\sqrt{6} - 4\sqrt{2})(3\sqrt{6} + \sqrt{2})\).The steps for binomial multiplication can be summarized as:
  • Identify the two binomials.
  • Multiply each term in the first binomial by each term in the second binomial.
  • Simplify each individual product.
  • Combine like terms.
By multiplying \(\sqrt{6}\) by \(3\sqrt{6}\) and \(\sqrt{2}\), and then doing the same with \(-4\sqrt{2}\), the expression results in four terms. Simplifying each leads to the expression \(10 - 22\sqrt{3}\).This example shows how binomial multiplication not only involves multiplying terms but also careful simplification to reach the final expression.

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Most popular questions from this chapter

Multiply. $$ \left(2 x^{1 / 3}+3\right)\left(2 x^{1 / 3}-3\right) $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(3 x^{1 / 4}\right)^{3}}{x^{1 / 12}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(m^{2} n\right)^{1 / 4}}{m^{-1 / 2} n^{5 / 8}} $$

Multiply. $$ y^{1 / 2}\left(y^{1 / 2}-y^{2 / 3}\right) $$

Simplify each exponential expression. $$\left(-2 x^{3} y^{2}\right)^{5}$$
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