Problem 23 - Precalculus

Question

Solve the equation or inequality. $$x^{\frac{2}{3}}=4$$

Step-by-Step Solution

Answer

The solution is \(x = 8\).

Step 1: Understand the Equation

The equation given is \(x^{\frac{2}{3}} = 4\). This equation means we are looking for a number \(x\) such that raising \(x\) to the power of \(\frac{2}{3}\) results in 4.

Step 2: Raise to a Reciprocal Power

To solve for \(x\), we can eliminate the fractional exponent by raising both sides to the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This gives us:\[(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}\]Thus, \(x\) becomes:\(x = 4^{\frac{3}{2}}\).

Step 3: Simplify the Right Side

Now, we calculate \(4^{\frac{3}{2}}\). This can be rewritten as \((4^{\frac{1}{2}})^3\), which means take the square root of 4 and then cube it. The square root of 4 is 2, so the expression becomes \(2^3\).

Step 4: Calculate the Final Value

Calculate \(2^3\):\[2^3 = 8\]Therefore, the value of \(x\) is 8.

Key Concepts

Understanding Fractional Exponents

Fractional exponents can be seen as a bridge between exponents and roots. When you have something like \(x^{\frac{2}{3}}\), it’s equivalent to taking the square of \(x\) and then finding the cube root of the result. Or, conversely, taking the cube root first and then squaring that result.

• The numerator of the fraction (in this case, 2) indicates the power to which the base number should be raised.
• The denominator (in this case, 3) tells you what root to take.

This means \(x^{\frac{2}{3}}\) can alternatively be expressed as \((x^2)^{\frac{1}{3}}\) or \((x^{\frac{1}{3}})^2\). Understanding this helps you perform complex operations by breaking them into more manageable steps: first handle the root or the power, then the inverse operation.

Solving Equations with Fractional Exponents

When solving equations with fractional exponents, the key is to eliminate the fractional power to simplify the equation to a form you can solve directly. In the exercise given (\(x^{\frac{2}{3}} = 4\)), we used the reciprocal of the exponent to remove the fraction.

• The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). Raising both sides of the equation by this power cancels out the fraction on \(x\).

So, \[(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}\]Simplifying the left side results in \(x\), while the right side needs further simplification, as shown in the problem solution. Getting rid of the fraction helps us to isolate \(x\) and solve the equation straightforwardly.

Applying Reciprocal Powers

Raising numbers to reciprocal powers is a smart way to "undo" a power raised to a fractional exponent. For instance, if given an equation like \(x^{\frac{2}{3}} = 4\), you can go the other route and transform it by raising to the power of \(\frac{3}{2}\).

• Remember: Raising a number to the power of its reciprocal cancels the effect of the initial exponent.

In practice:\[(x^{\frac{2}{3}})^{\frac{3}{2}} = x^{\frac{2}{3} \cdot \frac{3}{2}} = x^1 = x\]This results in \(x\), simplifying your solution process. On the right side, \(4^{\frac{3}{2}}\) is calculated separately by interpreting it first as taking the square root of 4, yielding 2, and then raising 2 to the third power, resulting in 8. By mastering these steps, you can tackle even more complex problems with confidence.