Question

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

Short answer

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

Step-by-step solution

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 \(x\).

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 \(x^3 = 216\).

Solve for x

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

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.

Key concepts

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.