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Chapter 9: Problem 171

In the following exercises, simplify. $$ 3 \sqrt{5 d}+8 \sqrt{5 d}-11 \sqrt{5 d} $$

Short Answer

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Step by step solution

01

Identify like terms

The terms in the expression are all like terms because they all contain the radical part \(\sqrt{5d}\).
02

Combine coefficients

Add the coefficients of \(\sqrt{5 d}\) together, which are 3, 8, and -11: \(3 + 8 - 11\).
03

Simplify the coefficients

Perform the arithmetic operation: \(3 + 8 = 11\), then \(11 - 11 = 0\).
04

Multiply coefficient by the radical

Since the sum of the coefficients is 0, multiply 0 by \(\sqrt{5d}\): \(0 \cdot \sqrt{5d} = 0\).

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Key Concepts

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

like terms
In algebra, 'like terms' refer to terms whose variables and their exponents are the same. For example, in the expression 3鈭�5d + 8鈭�5d - 11鈭�5d, all terms involve the same radical part 鈭�5d.
This makes them like terms, which can be combined.
Like terms simplify calculations by allowing addition or subtraction of their coefficients.
Identifying like terms is crucial for simplification, whether you鈥檙e dealing with regular variables or radicals.
combining coefficients
Once you've identified like terms, the next step is combining coefficients.
Coefficients are the numerical parts of the terms. In the original exercise, the coefficients are 3, 8, and -11 for the terms involving 鈭�5d.
To combine them, you perform simple arithmetic: 3 + 8 - 11.
The operation goes as follows:
  • Add 3 and 8, which equals 11
  • Subtract 11 from 11, leading to a final result of 0
This shows that the combined coefficient is 0.
Combining coefficients helps in simplifying expressions to their simplest form.
radicals
Radicals involve the notation 鈭�, indicating the square root of a number or expression.
Radicals can have coefficients and can sometimes be simplified themselves, but the radical part must remain the same to combine terms.
In the given expression, all radicals are 鈭�5d, making it possible to combine their coefficients.
Once the coefficients are combined and simplified to 0 in our example, the entire expression, 0 鈭�5d, simplifies to 0.
Knowing how to handle radicals is essential for simplifying many algebraic expressions, ensuring you can correctly combine and reduce terms.

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Most popular questions from this chapter

(a) Approximate \(\frac{1}{\sqrt{2}}\) by dividing \(\frac{1}{1.414}\) using long division without a calculator. (b) Rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) gives \(\frac{\sqrt{2}}{2}\). Approximate \(\frac{\sqrt{2}}{2}\) by dividing \(\frac{1.414}{2}\) using long division without a calculator. (C) Do you agree that rationalizing the denominator makes calculations easier? Why or why not?

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}} $$

In the following exercises, simplify. (a) \(\sqrt[3]{512}\) (b) \(\sqrt[4]{81}\) (c) \(\sqrt[5]{1}\)

(a) Simplify \(\sqrt{\frac{27}{3}}\) and explain all your steps. (b) Simplify \(\sqrt{\frac{27}{5}}\) and explain all your steps. (c) Why are the two methods of simplifying square roots different?

In the following exercises, simplify and rationalize the denominator. $$ -\frac{8}{3 \sqrt{6}} $$
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