Problem 84 - Algebra for College Students

Question

Solve each equation. $$x^{\frac{2}{3}}=2$$

Step-by-Step Solution

Answer

The solution to the equation is approximately 2.828.

Step 1: Understand the Equation

The given equation is \( x^{\frac{2}{3}} = 2 \). This means that when \( x \) is raised to the power of \( \frac{2}{3} \), the result equals 2.

Step 2: Isolate the Variable

To solve for \( x \), we need to eliminate the exponent. We can do this by raising both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). Thus, we have: \((x^{\frac{2}{3}})^{\frac{3}{2}} = 2^{\frac{3}{2}}\).

Step 3: Simplify the Left Side

On the left side of the equation, \((x^{\frac{2}{3}})^{\frac{3}{2}}\) simplifies to \( x^{1} \) because \( \frac{2}{3} \times \frac{3}{2} = 1 \). So, the equation becomes \( x = 2^{\frac{3}{2}} \).

Step 4: Simplify the Right Side

Calculate \( 2^{\frac{3}{2}} \). This expression means taking the square root of 2 and then cubing the result, or vice versa. \( 2^{\frac{1}{2}} \) is the square root of 2, which is approximately 1.414. Cubing this gives \( 2.828 \). Alternatively, you can find \( 2^{3} = 8 \) first and then take the square root, which also results in \( 2.828 \).

Step 5: Conclude the Solution

The solution to the equation \( x^{\frac{2}{3}} = 2 \) is \( x \approx 2.828 \).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base number is the number being multiplied by itself, while the exponent indicates how many times the base number is used as a factor. For example, in the expression \( x^3 \), the base is \( x \), and the exponent is 3, indicating that \( x \) is multiplied by itself three times, resulting in \( x \times x \times x \).

When dealing with an equation like \( x^{\frac{2}{3}} = 2 \), the exponent \( \frac{2}{3} \) implies that \( x \) is being raised to the power of \( \frac{2}{3} \), involving both a square and a cube root. Understanding exponentiation is crucial because it allows us to manipulate and solve equations by applying operations such as multiplication, division, and taking roots. In many problems, raising both sides of an equation to a reciprocal power helps to "undo" the exponentiation, helping to isolate the variable for easier solving.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division) that collectively represent a value. These can be as simple as \( 3x + 2 \), or more complex like \( x^{\frac{2}{3}} + 5x - 7 \). Expressions can often be manipulated using algebraic rules to simplify or solve them in the context of equations.

In the original exercise, we are dealing with the algebraic expression \( x^{\frac{2}{3}} = 2 \). In this case, one side of the equation is an algebraic expression that defines \( x \) raised to a specific power. To solve for \( x \) in an equation, we often rely on understanding how different parts of the expression relate, as well as how different algebraic rules, such as the properties of exponents, can be used to isolate variables and find their values.

- Always simplify expressions when possible to make solving easier.
- Use reverse operations to isolate the variable when solving equations.

Rational Exponents

Rational exponents involve exponents that are fractions, such as \( \frac{2}{3} \) in the expression \( x^{\frac{2}{3}} \). Rational exponents provide an alternative way to represent roots. The numerator of the fraction represents the power, while the denominator represents the root. For instance, \( x^{\frac{2}{3}} \) is equivalent to the cube root of \( x^2 \) or the square of the cube root of \( x \).

Handling rational exponents effectively involves understanding these translations between exponents and roots. When we solve an equation like \( x^{\frac{2}{3}} = 2 \), we raise both sides of the equation to the reciprocal exponent \( \frac{3}{2} \) to cancel out the initial rational exponent on \( x \). This manipulation turns our initial problem into finding \( x \) in simpler terms, showing how rational exponents can often simplify to more familiar exponentiation or root operations.

- Always convert complex rational exponents into step-by-step operations like roots and powers to simplify the solving process.
- Using the properties of exponents helps in restructuring and solving complex equations efficiently.