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Chapter 10: Problem 3

How do you know that a radical expression containing a cube root is completely simplified?

Short Answer

Expert verified
A radical expression containing a cube root is completely simplified when there are no factors with a power of three remaining under the radical. To check for simplification, factor the radicand into its prime factors, identify any factors with a power of three, and ensure that none are left in the radicand after simplification. For example, for the expression \(\sqrt[3]{24}\), after simplifying, it becomes \(2\sqrt[3]{3}\), and there are no factors with a power of three left in the radicand, so it's completely simplified.

Step by step solution

01

Identify the cube root expression

Locate the cube root expression in the problem. Recall that the cube root of a number is written as \(\sqrt[3]{x}\), where x is the radicand.
02

Factor the radicand

Determine the prime factorization of the radicand. Break down the radicand into a product of its prime factors. For example, if the radicand is 24, you would find that the prime factorization is \(2^3 * 3\).
03

Check for factors with a power of three

Analyze the prime factors obtained in Step 2, and look for any factors that appear three times (i.e., have a power of three). If a factor appears three or more times, it can be taken out of the cube root in the simplified expression. If there are no factors with a power of three, the expression is already completely simplified.
04

Ensure no further simplification is possible

Examine the expression to guarantee that it is completely simplified and has no remaining cube root factors. The expression is simplified completely when you cannot find any more factors that have a power of three within the radicand. For example, if we had the expression \(\sqrt[3]{24}\), after simplifying, it would become \(\sqrt[3]{2^3} * \sqrt[3]{3} = 2\sqrt[3]{3}\). Since there are no factors with a power of three left in the radicand, the expression is completely simplified.

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Most popular questions from this chapter

The following radical expressions do not have the same indices. Perform the indicated operation, and write the answer in simplest radical form. Assume the variables represent positive real numbers. $$\sqrt[5]{c^{3}} \cdot \sqrt[3]{c^{2}}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\frac{6}{\sqrt[3]{u}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{5}{4+\sqrt{6}}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[3]{\frac{3}{n^{2}}}$$

Find the conjugate of each binomial. Then, multiply the binomial by its conjugate. $$(\sqrt{2}+\sqrt{6})$$
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