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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 8
Chapter 5, Problem 8

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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Solve each equation. Round answers to the nearest hundredth as needed. (1/4)x=64

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Textbook Question

Answer each of the following. Between what two consecutive integers must log2 12 lie?

Textbook Question

Determine whether each function graphed or defined is one-to-one.

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. log 1012

Textbook Question

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. 34 = 81

Textbook Question

Answer each of the following. Write log3 12 in terms of natural logarithms using the change-of-base theorem.