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

검증된 단계별 안내

Recognize that the equation is \(x^{\frac{2}{3}} = 36\). Our goal is to solve for \(x\).

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 \(\left(x^{\frac{2}{3}}\right)^{\frac{3}{2}} = 36^{\frac{3}{2}}\).

Simplify the left side using the property of exponents: \(\left(a^{m}\right)^{n} = a^{mn}\). So, \(x^{\frac{2}{3} \times \frac{3}{2}} = x^{1} = x\).

Now, focus on simplifying the right side: \(36^{\frac{3}{2}}\). Recall that \(a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}\). So, \(36^{\frac{3}{2}} = \left(\sqrt{36}\right)^{3}\).

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Solve each equation. Round answers to the nearest hundredth as needed. x2/3 =36