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Chapter 10: Problem 23

What formula can be used to multiply \((5+\sqrt{6})(5-\sqrt{6}) ?\)

Short Answer

Expert verified
The multiplication $(5+\sqrt{6})(5-\sqrt{6})$ can be simplified using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). In this case, \(a = 5\) and \(b = \sqrt{6}\). Applying the formula, we get: \((5+\sqrt{6})(5-\sqrt{6}) = 5^2 - (\sqrt{6})^2 = 25 - 6 = 19\).

Step by step solution

01

Identify the expressions as difference of squares

We are given the expressions \((5+\sqrt{6})(5-\sqrt{6})\). To simplify these expressions, notice that they can be written in the form \((a+b)(a-b)\), which is the difference of squares pattern (here, \(a=5\) and \(b=\sqrt{6}\)).
02

Apply the difference of squares formula

The difference of squares formula states that \((a+b)(a-b) = a^2 - b^2\). In our case, \(a = 5\) and \(b = \sqrt{6}\). Applying this formula to our given expressions, we get: \((5+\sqrt{6})(5-\sqrt{6}) = 5^2 - (\sqrt{6})^2\)
03

Simplify the expressions

Now, we will simplify the expression: \(5^2 - (\sqrt{6})^2 = 25 - 6\) \(25 - 6 = 19\) So, \((5+\sqrt{6})(5-\sqrt{6})=19\).

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Most popular questions from this chapter

In your own words, explain how to rationalize the denominator of an expression containing one term in the denominator.

Fill in the blank. Assume all variables represent positive real numbers. $$\sqrt[5]{16} \cdot \sqrt[5]{7}=\sqrt[5]{2^{5}}=2$$

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