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Chapter 0: Problem 19

evaluate the expression for the given value of x. $$ \sqrt[3]{x} \quad x=-154 $$

Short Answer

Expert verified
The value of the expression \(\sqrt[3]{-154}\) is approximately -5.37.

Step by step solution

01

Understand the expression

The expression given is \(\sqrt[3]{x}\), which denotes the cube root of \(x\). This means we are asked to find a number that, when cubed (the number multiplied by itself twice), results in \(x\).
02

Substitute the value of x

We are given that \(x = -154\). So, we substitute \(-154\) for \(x\) in the expression \(\sqrt[3]{x}\), which gives \(\sqrt[3]{-154}\).
03

Evaluate the cube root

Finally, using a calculator or cube root table, the cubic root of -154 is approximately -5.37 when rounded to the nearest hundredth.

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Key Concepts

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

Evaluating Expressions
Evaluating expressions is a fundamental skill in mathematics involving the calculation of a numerical value from a given mathematical statement. When an expression is given, it often contains variables and constants along with mathematical operations such as addition, subtraction, multiplication, division, or roots.
  • The primary goal is to determine the numerical value the expression represents.
  • To start, identify the operations and the values involved.
  • Perform the operations using an appropriate order, such as following the Order of Operations: Parentheses, Exponents (roots), Multiplication and Division (left to right), Addition and Subtraction (left to right).
  • Expressions are solved step-by-step to ensure accuracy in the outcome.
This approach helps in understanding where to begin and how to progress through the problem logically.
Substitution in Expressions
Substitution involves replacing a variable in a mathematical expression with a specific numerical value. This is crucial when the expression includes variables, and a particular value is given for solving purposes.

  • To use substitution, identify the variable in the expression and the value given for it, such as in this exercise where \( x = -154 \).
  • Replacing the variable with the given value transforms the abstract expression into a concrete numerical form, making it easier to evaluate.
  • After substitution, recalculate the expression using the substituted values to achieve a definitive answer.
This process clarifies the expression and eliminates the variable, allowing for straightforward computations.
Negative Values in Cube Roots
Cube roots of negative numbers might seem intimidating at first, but they follow consistent rules that make them comprehensible. Unlike square roots, cube roots can be negative since cubing a negative number results in a negative product.

  • Understanding that the cube root of a negative number, like \( \sqrt[3]{-154} \), is valid is essential.
  • Cube roots seek a number that, when multiplied by itself three times (cubed), produces the negative number in question.
  • For instance, to find \( \sqrt[3]{-154} \), the objective is to determine a value that, when cubed, equals -154.
  • This can typically be approximated using a calculator; for our case, it's about -5.37 when rounded to the nearest hundredth.
Recognizing that cube roots apply to both positive and negative numbers helps simplify various mathematical problems.

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