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Chapter 8: Problem 14

Simplify. (a) \(\sqrt[3]{125}\) (b) \(\sqrt[4]{1296}\) (c)\(\sqrt[5]{1024}\)

Short Answer

Expert verified
a) 5, b) 6, c) 4

Step by step solution

01

Simplify (a)

To simplify \(\backslash sqrt[3]{125}\), recognize that 125 is a perfect cube. 125 can be written as \(5^3\). Therefore, \(\backslash sqrt[3]{125} = \backslash sqrt[3]{5^3} = 5\).
02

Simplify (b)

To simplify \(\backslash sqrt[4]{1296}\), recognize that 1296 is a perfect fourth power. 1296 can be written as \(6^4\). Therefore, \(\backslash sqrt[4]{1296} = \backslash sqrt[4]{6^4} = 6\).
03

Simplify (c)

To simplify \(\backslash sqrt[5]{1024}\), recognize that 1024 is a perfect fifth power. 1024 can be written as \(4^5\). Therefore, \(\backlash sqrt[5]{1024} = \backslash sqrt[5]{4^5} = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Root
Understanding the cube root is crucial when simplifying radical expressions. The cube root of a number, expressed as \sqrt[3]{a}\, is a value that, when multiplied by itself three times, equals **a**. Consider the expression \sqrt[3]{125}\. To simplify it, note that 125 can be expressed as \(5^3\). Therefore, \sqrt[3]{125} = \sqrt[3]{5^3} = 5\.

Here's the process step-by-step:
  • Recognize that 125 is a perfect cube.
  • Write 125 as \(5^3\).
  • Apply the cube root, resulting in 5.

This principle can be applied to any number that is a perfect cube. By identifying the base number and its exponent, you can quickly simplify cube roots.
Fourth Root
The fourth root of a number is found by determining what value, when raised to the power of **four** equals the original number. Written as \sqrt[4]{a}\, this is needed for simplifying certain radical expressions. Consider the given example \sqrt[4]{1296}\. We recognize that 1296 is a perfect fourth power and can be expressed as \(6^4\). Therefore, \sqrt[4]{1296} = \sqrt[4]{6^4} = 6\.

Follow these steps to simplify:
  • Identify that 1296 is a perfect fourth power.
  • Express 1296 as \(6^4\).
  • Simplify by applying the fourth root, giving you 6.

Identifying perfect fourth powers and their roots simplifies the otherwise complex-looking expressions.
Fifth Root
Fifth roots are simplified similarly to fourth and cube roots. The fifth root of a number is a value that, when raised to the power of **five** equals the original number. It is represented as \sqrt[5]{a}\. To simplify \sqrt[5]{1024}\, realize that 1024 can be written as \(4^5\). Therefore, \sqrt[5]{1024} = \sqrt[5]{4^5} = 4\.

Here's the simplification process:
  • Determine that 1024 is a perfect fifth power.
  • Express 1024 as \(4^5\).
  • Take the fifth root to simplify, resulting in 4.

Recognizing and converting numbers into their perfect powers makes solving fifth roots straightforward.

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