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Chapter 4: Problem 3

If \(x=y=z\) and \(x y z=216, x\) is: A. 72 B. 9 C. 12 D 6 E. 4

Short Answer

Expert verified
The value of \(x\) is \textbf{6}\).

Step by step solution

01

Analyze the given information

It is given that \(x = y = z\) which means all three variables are equal. Also, \(x \cdot y \cdot z = 216\). We need to find the value of \(x\).
02

Substitute the equal variables

Since \(x = y = z\), substitute \(x\) for \(y\) and \(z\) in the equation. Therefore, \(x \cdot x \cdot x = 216\), which simplifies to \(x^3 = 216\).
03

Solve for x

To find \(x\), take the cube root of both sides of the equation: \(\root{3}{x^3} = \root{3}{216}\). Therefore, \(x = 6\).
04

Verify the solution

Verify by cubing \(x = 6\). \( 6^3 = 6 \cdot 6 \cdot 6 = 216\), which matches the given condition. Hence, the solution is correct.

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Key Concepts

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

Cube Roots
Cube roots are the opposite of cubing a number, which means they undo the operation of raising a number to the power of three. If you have an expression like \(x^3 = 216\), you can find \(x\) by figuring out the cube root of 216. The cube root of a number \(N\), written as \( \root{3}{N} \), is the value that, when multiplied by itself three times, gives \(N\). In our case, \( \root{3}{216} = 6\) because \(6 \times 6 \times 6 = 216 \).

Generally, cube roots can be found using a calculator or by recognizing common cubes:
  • \( \root{3}{1} = 1 \)
  • \( \root{3}{8} = 2 \)
  • \( \root{3}{27} = 3 \)
and so on. These basics will help you identify cube roots quickly.
Equations
An equation is a statement that asserts the equality of two expressions. In the given problem, we had the equation \(x \times y \times z = 216\). Since we were told that \(x = y = z\), we substituted all variables with \(x\), turning the equation into \(x \times x \times x = 216\) or \(x^3 = 216\).

The goal when solving an equation is to isolate the variable you are solving for. Here, we isolated \(x\) by taking the cube root of both sides of the equation, thus solving \(x^3 = 216\) by finding \( \root{3}{216}\). This technique uses the properties of equality and operations to maintain balance on both sides of the equation.
Algebra
Algebra involves the use of symbols and letters to represent numbers and quantities in formulas and equations. In our problem, we used algebra to represent three unknowns with the same variable \(x\), simplifying our equation to \(x^3 = 216\).

One fundamental concept in algebra is substitution, which allows us to replace variables with actual values or other variables. This makes the equations simpler and more manageable. In the given problem, substitution and recognizing equal variables allowed for an easier solution path.

Mastery of algebraic manipulation is essential. This includes
  • Solving for unknowns
  • Using properties of operations, like taking roots
  • Understanding variable relationships and their balances.
Grasping these concepts will help you solve more complex algebra problems with confidence.

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Most popular questions from this chapter

Historically, an increase in the level of production virtually always leads to a reduction in unit cost. A. virtually always leads to a reduction B. are going to lead to a reduction C. would lead to a reduction D. mustn't lead to a reduction E. may lead to a reduction

If you walk around a square that covers an area of \(225 \mathrm{~m}^2\), how far would you walk? A. \(15 \mathrm{~m}\) B. \(20 \mathrm{~m}\) C. \(25 \mathrm{~m}\) D. \(30 \mathrm{~m}\) E. \(60 \mathrm{~m}\)

What is the common factor that two numbers have if one of the numbers is 5 times the other and their sum is 30 ? A. 2 B. 3 C. 5 D. 7 E. 9

Is \(x\) positive? (1) \(x^2-9=0\) (2) \(x^3-27=0\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. \(\quad 1\) and 2 together are not sufficient

The rooms were full of bottles and each bottle was filled with water. A. The rooms were full of bottles and each bottle was filled with water. B. The rooms were full of bottles and each was filled with water. C. The rooms were full with bottles and each bottle was filled with water. D. The rooms were full of bottles and each was full of water. E. The rooms were full of bottles and each was filled of water.
See all solutions

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