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Chapter 8: Problem 307

Solve. \(\sqrt[3]{4 x+5}-2=-5\)

Short Answer

Expert verified
The solution is x = -8.

Step by step solution

01

Isolate the Cube Root

Add 2 to both sides of the equation to isolate the cube root expression \[\sqrt[3]{4x + 5} = -3\]
02

Eliminate the Cube Root

Cube both sides of the equation to eliminate the cube root \[(\sqrt[3]{4x + 5})^3 = (-3)^3\]which simplifies to \[4x + 5 = -27\]
03

Solve for x

Subtract 5 from both sides to isolate the variable term \[4x = -32\]Then, divide both sides by 4 \[x = -8\]

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Key Concepts

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

Cube Roots
A cube root is a number that, when multiplied by itself three times, gives the original number. In symbols, the cube root of a number y is written as \( \sqrt[3]{y} \). For example, \(\sqrt[3]{27} = 3 \) because \(3 \times 3 \times 3 = 27\).
When solving equations with cube roots, one of the first steps is usually to eliminate the cube root by cubing both sides of the equation. This makes it easier to solve for the variable.
Isolating Terms
To solve equations, we need to isolate the variable term on one side of the equation. We can do this by performing the same mathematical operations on both sides of the equation. For instance, in the given exercise, we start by isolating the cube root term \( \sqrt[3]{4 x+5} \). To do that, we add 2 to both sides of the original equation:
\(\sqrt[3]{4 x+5} - 2 + 2 = -5 + 2\)
Which simplifies to:

\(\sqrt[3]{4 x+5} = -3\)\.
Isolating terms helps us simplify the equation step-by-step.
Variable Isolation
Variable isolation refers to arranging the equation so that the variable (usually x) is by itself on one side of the equation. This makes it easier to solve for that variable. We often have to perform various operations like adding, subtracting, multiplying, or dividing both sides of the equation to achieve this.
In this case, after eliminating the cube root, we had
\(4x + 5 = -27 \).
To isolate the variable term \[4x\], we subtract 5 from both sides. This gives us \(4x = -32\)
This is crucial for the next step, solving for the variable x.
Solving for x
Once the variable term is isolated, the final step is to solve for x. In this example, we have \(4x = -32\)
To find the value of x, we divide both sides by 4:
\(x = \-32 \div 4 \)
and this simplifies to
\(x = -8\).
As you can see, solving for x is straightforward once the variable term is isolated. This method can be used for different types of algebraic equations to find the value of the variable.

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