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Chapter 10: Problem 4

Add or subtract as indicated. $$9 \sqrt[3]{7}-4 \sqrt[3]{7}$$

Short Answer

Expert verified
\(5 \sqrt[3]{7}\)

Step by step solution

01

Identify like terms

The like terms given in the exercise are \(9 \sqrt[3]{7}\) and \(4 \sqrt[3]{7}\).
02

Subtract the like terms

Subtract the like terms by keeping the common factor same i.e., \(\sqrt[3]{7}\), and subtract the numbers in front of it i.e., 9 and 4. So, \(9 \sqrt[3]{7} - 4 \sqrt[3]{7} = (9-4) \sqrt[3]{7}\)
03

Simplify the Solution

Simplify the expression \( (9-4) \sqrt[3]{7}\) giving us \(5 \sqrt[3]{7}\)

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Key Concepts

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

Like terms
When dealing with expressions that involve radicals, it's important to understand the concept of like terms. Like terms are terms that have the same radical part. In this specific exercise, both terms are associated with the cube root of 7, i.e., \(\sqrt[3]{7}\). This is what makes them like terms.

For terms to be like terms, they must have exactly the same index and radicand. The index is the small number outside the radical symbol that indicates the root (in this case, 3 for cube roots), and the radicand is the number or expression inside the radical (here, 7).
  • Make sure both terms inside the expression have the same root and radicand before combining them.
  • Only like terms can be simplified using operations like addition and subtraction.
Recognizing like terms in radical expressions is crucial as it allows you to simplify the expressions effectively.
Simplifying expressions
Simplifying expressions in mathematics involves reducing them to their simplest form for easier computation and understanding. In the context of expressions involving radicals, once you identify like terms, you can simplify them.
  • Identify common factors, such as the radical part \(\sqrt[3]{7}\) in our exercise.
  • Combine the coefficients (numbers in front of the radicals) of these like terms.
In the exercise, the expression simplifies by performing arithmetic on the coefficients. You subtract 4 from 9 because both are coefficients of like terms, giving you 5.

The simplified expression is then \(5 \sqrt[3]{7}\). This process makes complex-looking expressions much less daunting, making further mathematical operations easier.
Cube roots
Cube roots help us find the value that, when multiplied by itself three times, results in the given number. Cube roots are represented as \(\sqrt[3]{x}\), where \(x\) is the number for which we want to find the cube root.

In expressions like those in our exercise, understanding cube roots play a key role.
  • The cube root 3 indicates that we are seeking a value that when raised to power 3 equals the radicand, such as 7 in the given expression \(\sqrt[3]{7}\).
  • Cube roots can be positive or negative, as both \(x\) and \(-x\) can satisfy the cube requirement for many numbers.
Dealing with cube roots often involves recognizing them as components of terms in expressions. Understanding how they work helps in adding, subtracting, or further manipulating these terms accurately in equations.

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Most popular questions from this chapter

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\frac{3}{\sqrt[4]{x^{5} y^{3}}}$$

In Exercises \(137-140\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Some irrational numbers are not complex numbers.

Solve each equation. $$\sqrt{\sqrt{x}+\sqrt{x+9}}=3$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{15}{\sqrt{6}+1}$$

Will help you prepare for the material covered in the next section. Rationalize the denominator: \(\frac{7+4 \sqrt{2}}{2-5 \sqrt{2}}\)
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