Question

What is the value of \(x\) ? (1) \(x^3=8\) (2) \(x^2+2=6\) 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. 1 and 2 together are not sufficient

Short answer

A. 1 alone, not 2 alone

Step-by-step solution

- Analyze Equation (1)

Consider the equation: \[ x^3 = 8 \]To find the value of \( x \), take the cube root of both sides: \[ x = \sqrt[3]{8} \]\[ x = 2 \]This means that the value of \( x \) is 2 when solved using Equation (1).

- Verify Value with Equation (2)

Now, let's consider Equation (2): \[ x^2 + 2 = 6 \]Subtract 2 from both sides to isolate \( x^2 \): \[ x^2 = 4 \]To find the value of \( x \), take the square root of both sides: \[ x = \sqrt{4} \]\[ x = 2 \] or \[ x = -2 \]Since we have already found that \( x = 2 \) from Equation (1), we verify that 2 is still a valid solution for Equation (2). However, Equation (2) alone can give \( x = -2 \) which makes it different.

- Determine the Sufficiency

Based on the solutions above, Equation (1) alone is sufficient to determine the value of \( x \) as 2. However, Equation (2) alone could lead to two possible values of \( x \) (2 and -2). Hence, Equation (1) is sufficient while Equation (2) alone is not.

Key concepts

cube root

The cube root is a vital mathematical concept used to determine what number, when multiplied by itself three times, will equal a given value. In this problem, we are given the equation \( x^3 = 8 \). To solve for \( x \), we take the cube root of both sides. The cube root of 8 is calculated as follows: \( x = \sqrt[3]{8} \), which equals 2. Therefore, \( x = 2 \). Cube roots are especially useful in GMAT problems that involve volumes and other three-dimensional calculations.

square root

A square root is another essential mathematical concept that indicates what number, when multiplied by itself, equals the given number. In the given problem, the equation \( x^2 + 2 = 6 \), after simplification, results in \( x^2 = 4 \). To isolate \( x \), we take the square root of 4, resulting in two possible values: \( x = \sqrt{4} \), which includes both 2 and -2. Therefore, we find \( x \) could be either 2 or -2. Square roots frequently appear on the GMAT in the context of geometry, quadratic equations, and other algebraic manipulations.

equation solving

Equation solving is a fundamental skill in algebra and GMAT preparation. It involves manipulating the equation to isolate the unknown variable. For instance, in the equation \( x^3 = 8 \), solving for \( x \) requires taking the cube root. For \( x^2 + 2 = 6 \), solving involves isolating \( x^2 \) by subtracting 2 from both sides and then taking the square root. Mastery in solving different types of equations, such as linear, quadratic, and cubic, is crucial for success in the math section of standardized tests like the GMAT.

algebra

Algebra forms the bedrock of most mathematical problem solving, especially in standardized tests like the GMAT. This branch of mathematics involves the manipulation of symbols and variables to solve equations. Key concepts such as cube roots, square roots, and isolating variables are all part of algebra. In this exercise, we use algebraic principles to find the value of \( x \). For \( x^3 = 8 \), applying the cube root provides us with \( x = 2 \). For \( x^2 + 2 = 6 \), we solve it by isolating \( x^2 \), resulting in \( x = 2 \) or \( x = -2 \). Understanding algebra is critical for solving real-world problems and excelling in the GMAT.

GMAT preparation

Effective GMAT preparation requires a strong foundation in core mathematical concepts such as algebra, equation solving, and understanding roots. A well-structured study plan should include:

• Reviewing key math concepts frequently tested on the GMAT
• Practicing problem-solving skills through sample questions and mock exams
• Focusing on time management to answer questions efficiently within the test limits

By mastering these areas, students will be better equipped to tackle a variety of problems, including ones involving cube roots and square roots, which often appear in GMAT math sections.
Sufficient practice and a clear understanding of mathematical principles will boost confidence and performance during the exam.