Problem 82 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 2\sqrt{2}\).

Step 1: Understand the Equation

The equation given is \(x^{\frac{2}{3}} = 2\). This is an exponential equation where the variable \(x\) is raised to the power of \(\frac{2}{3}\). Our goal is to solve for \(x\).

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 power of \(\frac{3}{2}\), which is the reciprocal of \(\frac{2}{3}\). This gives us \(\left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 2^{\frac{3}{2}}\).

Step 3: Simplify the Expression

Raising a power to another power means multiplying the exponents. Thus, \((x^{\frac{2}{3}})^{\frac{3}{2}} = x^{(\frac{2}{3} \cdot \frac{3}{2})} = x^1 = x\). So the left side simplifies to \(x\).

Step 4: Calculate the Right Side

We now need to calculate \(2^{\frac{3}{2}}\). This can be rewritten as \((2^3)^{\frac{1}{2}}\), which is equal to \(\sqrt{8}\). Since \(8\) is \(2\times2\times2\), \(\sqrt{8} = 2\sqrt{2}\).

Step 5: Solve for the Solution

After solving both sides, we see that \(x = 2^{\frac{3}{2}} = 2\sqrt{2}\), which is approximately \(2.828\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the given equation, \(x^{\frac{2}{3}} = 2\), our task is to solve for \(x\). This means determining what number, when raised to the power \(\frac{2}{3}\), results in \(2\).

To solve, we need to "undo" the exponent. We start by raising both sides of the equation to a power that cancels out the original exponent. Specifically, we use the reciprocal power. For \(\frac{2}{3}\), the reciprocal is \(\frac{3}{2}\).

• This means we apply \(\left(\cdot\right)^{\frac{3}{2}}\) to both sides, effectively isolating \(x\). For instance, \(\left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 2^{\frac{3}{2}}\).
• After simplification, \(x\) remains on one side, and \(x = 2^{\frac{3}{2}}\).

The objective in solving equations is to manipulate them until the variable stands alone on one side, telling us its value.

Rational Exponents

Rational exponents are a way of expressing roots and powers in a unified form. A rational exponent like \(\frac{2}{3}\) signifies that the base is both rooted and raised to a power. For instance, in \(x^{\frac{2}{3}}\), \(x\) is squared (raised to the power 2) and then the result is taken to the cube root.

Understanding rational exponents is crucial to solving equations of this type. Breaking down the exponent \(\frac{2}{3}\):

• The denominator \(3\) indicates the type of root (cube root in this case).
• The numerator \(2\) represents the power to which the base is raised.

When converting roots to rational exponents, remember that \(\sqrt[3]{x^2} = x^{\frac{2}{3}}\). This aspect of algebraic manipulation allows us to handle roots and powers consistently, simplifying calculations and transformations.

Algebraic Manipulation

Algebraic manipulation refers to the use of algebraic rules and properties to transform an equation into a simpler form or to solve it. Key techniques include the use of reciprocal powers, distribution, and factoring.

For the equation \(x^{\frac{2}{3}} = 2\), we employed reciprocal exponents to manipulate and simplify the expression.

• Raising both sides to the power \(\frac{3}{2}\) exploited the power rule \((a^m)^n = a^{mn}\), which simplifies our equation perfectly to \(x = 2^{\frac{3}{2}}\).
• This made use of the rule that when the exponent product \(\frac{2}{3} \cdot \frac{3}{2} = 1\), the base \(x\) is left in its simplest form.

Calculating \(2^{\frac{3}{2}}\) involves rewriting it in terms of radicals and familiar fractions. Recognizing that \(2^{\frac{3}{2}} = \sqrt{8} = 2\sqrt{2}\), the problem simplifies, showcasing the power of algebraic manipulation in solving equations.