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Chapter 10: Problem 33

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$(\sqrt{11}+3)(\sqrt{11}-6)$$

Short Answer

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Step by step solution

01

Apply the Distributive Property

To simplify ewline (ewline sqrtewline {11ewline }+ ewline 3ewline )(ewline sqrtewline {11ewline }鈥揺wline 6ewline ),first apply the distributive property (also known as the FOIL method for binomials). Multiply each term in the first binomial by each term in the second binomial.This yields: ewline (ewline sqrtewline {11})ewline (ewline sqrtewline {11}ewline ) 鈥� )(ewline sqrtewline {11}ewline )(ewline -ewline 6ewline ) + (ewline 3ewline )(ewline sqrtewline {11}ewline ) 鈥� (ewline 3ewline )( -ewline 6ewline )
02

Multiply the Terms

Perform the multiplication: ewline 1.(ewline sqrtewline {11ewline })(ewline sqrtewline {11}ewline ) ewline =11 2. (ewline sqrtewline {11})(ewline -6ewline ) = -ewline 6ewline sqrtewline {11} 3.(ewline 3ewline )(ewline sqrtewline {11}) = +ewline 3ewline sqrtewline {11} 4. (ewline 3ewline )( -ewline 6ewline ) = -1ewline 8Thus, we have 11 鈥揺wline 6ewline sqrtewline +ewline|3ewline sqrtewline {11}ewline {11}-18
03

Simplify the Expression

Combine the radical terms: -ewline 16 +ewline 3 +ewline sqrtewline {11} becomes ewline 11 -18= -7de11 Hence , the simplified result is -7.

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Key Concepts

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

distributive property
The distributive property is a core principle in algebra that helps simplify expressions. It states that for any numbers or expressions, when you multiply a sum by a number, you distribute the multiplication over each addend within the parentheses. For example, in general terms, it follows the formula:


. Let's apply this to our problem: \((\begin{bmatrix} \begin{Vmatrix} sqrt [11 + 3 x sqrt) \begin{bmatrix} sqrt) {11_{x-6}) ] \rightarrow \begin{bmatrix} sqrtebt\begin ewline ewline({3}(begin{bmatrix}{(sqrt) ewlineote df12 {sqd}(3_{3} {soln}
This yields:
  • \)\begin{aewlineewlineewlineewline}\(11 - 6\)ewline{comn \)ewlineewline\( Line2(YXewline{b.)\beginewline3)\)NEW
    • ewline
  • M2 3^6 Notesend, disclose ewlineNow that you haveewline Distributive property, always correct and simplify answer.>.

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Most popular questions from this chapter

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$6 \sqrt{8}-\sqrt{98}$$

Algebraically. Be sure to label what the variable represents. Lynette can run at the rate of \(10 \mathrm{kph}\) and bike at the rate of \(24 \mathrm{kph}\). If she runs for one-half hour longer than she bikes and she covers a total of \(56 \mathrm{km},\) how long was she biking?

Use a calculator to find the following square roots. Round off your answers to the nearest hundredth. $$\sqrt{0.00049}$$

Use a calculator to find the following square roots. Round off your answers to the nearest hundredth. $$\sqrt{7.85}$$

Rationalize the denominators and simplify. $$\frac{12}{\sqrt{15}+\sqrt{3}}-\frac{30}{\sqrt{15}}$$
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