Solve each equation. Round answers to the nearest hundredth as needed. x^2/3 =36

Verified step by step guidance

1. Recognize that the equation is \(x^{\frac{2}{3}} = 36\). Our goal is to solve for \(x\).
2. To isolate \(x\), raise both sides of the equation to the reciprocal power of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This gives us \(\left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 36^{\frac{3}{2}}\).
3. Simplify the left side using the property of exponents: \(\left(a^{m}\right)^{n} = a^{mn}\). So, \(x^{\frac{2}{3} \times \frac{3}{2}} = x^{1} = x\).
4. Now, focus on simplifying the right side: \(36^{\frac{3}{2}}\). Recall that \(a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}\). So, \(36^{\frac{3}{2}} = \left(\sqrt{36}\right)^{3}\).
5. Calculate \(\sqrt{36}\), then cube the result to find the value of \(x\). Finally, consider both the positive and negative roots if applicable, and round your answer to the nearest hundredth.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers combined. For example, x^(2/3) means the cube root of x squared, or (x^(1/3))^2. Understanding how to manipulate and interpret these exponents is essential for solving equations involving fractional powers.

Isolating the Variable

To solve equations, isolate the variable by performing inverse operations. For x^(2/3) = 36, you can raise both sides to the reciprocal power (3/2) to undo the fractional exponent, allowing you to solve for x directly.

Rounding and Approximation

When solutions are irrational or decimals, rounding to a specified place value is necessary. Here, answers should be rounded to the nearest hundredth, meaning two decimal places, to provide a clear and practical numerical solution.