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Chapter 1: Problem 179

Will help you prepare for the material covered in the next section. If \(-8\) is substituted for \(x\) in the equation \(5 x^{\frac{2}{3}}+11 x^{\frac{1}{3}}+2=0\) is the resulting statement true or false?

Short Answer

Expert verified
The resulting statement is true.

Step by step solution

01

Substitute the x value

Replace \(x\) with \(-8\) in the initial equation, this gives us:\[5(-8)^{\frac{2}{3}} + 11(-8)^{\frac{1}{3}} + 2 = 0\]
02

Simplify equation

Now we simplify the equation. The cube root of \(-8\) is \(-2\), and \(-2\) squared gives us \(4\), therefore:\[5(4) + 11(-2) + 2 = 0\]Solving this equation gives us \(20 - 22 + 2 = 0\)
03

Evaluate the statement

After the final simplification, we get \(0 = 0\), which is a true statement.

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Key Concepts

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

Equation Solving
In mathematics, solving equations involves finding the unknown values that make a given equation true. An equation is a statement that tells us two expressions are equal. For example, in the equation given in this exercise, we want to know if substituting \(x = -8\) results in a true statement. The key to solving such equations is carefully following steps, like substitution and simplification.
  • Substitution: Replacing variables with specific values is crucial. Here, we substitute \(x = -8\) into the equation \(5x^{\frac{2}{3}} + 11x^{\frac{1}{3}} + 2 = 0\).

  • Simplification: Break down the equation into simpler parts to make it easier to solve. Calculate powers and roots, then perform arithmetic operations to see if both sides of the equation match.
If, after substitution and simplification, both sides of the equation are equal, the equation is verified. This systematic approach ensures that we handle equations accurately and efficiently.
Cube Roots
A cube root of a number is a value that, when multiplied by itself three times, results in the original number. For instance, the cube root of \(-8\) is \(-2\), because \(-2 \, \times \, -2 \, \times \, -2 = -8\). Cube roots are seen when working with rational exponents, as in our exercise with exponents like \({\frac{1}{3}}\), which denote taking the cube root.
  • Understanding Cube Roots: Knowing that the cube root of \(-8\) is \(-2\) is crucial for solving expressions like \(11(-8)^{\frac{1}{3}}\).

  • Negative Values: Remember that cube roots of negative numbers remain negative, unlike square roots which become imaginary. This distinction helps in simplifying expressions accurately.
Mastering cube roots allows us to comfortably manage parts of an equation that are not immediately clear, simplifying our problem-solving process significantly.
Simplifying Expressions
Simplifying expressions involves breaking down complex mathematical expressions into simpler forms. The goal is to make them easier to work with while preserving their value. This process includes dealing with exponents, roots, and arithmetic operations, as seen in this exercise.
  • Exponents and Roots: Rational exponents like \(\frac{2}{3}\) require understanding of power and root operations, which are combined here. Simplifying can involve finding cube roots and then squaring the result, such as transforming \(5(-8)^{\frac{2}{3}}\) into \(5(4)\).

  • Arithmetic Operations: After handling exponents and roots, simplify step-by-step through addition, subtraction, and other operations. In the original expression, combining \(5(4) + 11(-2) + 2\) efficiently leads us to the conclusion.
By simplifying expressions step-by-step, we reduce the chance of errors and obtain a clearer view of the solution path, guiding us safely toward the correct answer.

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Most popular questions from this chapter

Write a quadratic equation in general form whose solution set is \(\\{-3,5\\}\).

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In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90. a. What must you get on the final to earn an \(A\) in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

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