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Chapter 8: Problem 12

Simplify. (a) \(\sqrt[3]{27}\) (b) \(\sqrt[4]{16}\) (c) \(\sqrt[5]{243}\)

Short Answer

Expert verified
(a) 3, (b) 2, (c) 3

Step by step solution

01

Simplify \(\sqrt[3]{27}\)

To simplify \(\sqrt[3]{27}\), find the number that, when cubed, equals 27. \(\sqrt[3]{27} = 3\) because \(\3^3 = 27\).
02

Simplify \(\sqrt[4]{16}\)

To simplify \(\sqrt[4]{16}\), find the number that, when raised to the fourth power, equals 16. \(\sqrt[4]{16} = 2\) because \(\2^4 = 16\).
03

Simplify \(\sqrt[5]{243}\)

To simplify \(\sqrt[5]{243}\), find the number that, when raised to the fifth power, equals 243. \(\sqrt[5]{243} = 3\) because \(\3^5 = 243\).

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Key Concepts

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

cube root
The cube root is a special type of radical used to determine which number, when multiplied by itself three times (cubed), will give the original number. For example, when you see \(\root[3]{27}\), it is asking, 'What number, when multiplied by itself three times, equals 27?' In this case, the answer is 3 because \[3 \times 3 \times 3 = 27.\]To simplify cube roots:
  • Identify the number inside the cube root symbol.
  • Find the number that, when cubed, equals the original number.
Practicing simpler cubes can solidify your understanding, such as \(\root[3]{8} = 2\) since \[2^3 = 8.\].
fourth root
A fourth root is the number that, when multiplied by itself four times, equals the original number. For example, in the expression \(\root[4]{16}\), you need to find a number which when raised to the power of 4, results in 16. The correct number here is 2 because \[2 \times 2 \times 2 \times 2 = 16.\]To simplify fourth roots:
  • Identify the number inside the fourth root symbol.
  • Find the number that, when raised to the fourth power, equals the original number.
Other examples like \(\root[4]{81} = 3\) because \[3^4 = 81.\] can further help in practice.
fifth root
The fifth root represents the number that, when multiplied by itself five times, equals the original number. For instance, \(\root[5]{243}\) asks you to find what number must be raised to the power of 5 to get 243. Here, the number is 3 because \[3 \times 3 \times 3 \times 3 \times 3 = 243.\]To simplify fifth roots:
  • Identify the number inside the fifth root symbol.
  • Find the number that, when raised to the fifth power, equals the original number.
Other fifth root examples include \(\root[5]{32} = 2\) because \[2^5 = 32.\]
radicals
Radicals involve expressions that contain roots, such as square roots, cube roots, etc. Simplifying radicals means finding the root of a number and expressing it in its simplest form. For example, the square root of 9 is 3 because \[3^2 = 9.\]Here is how you can simplify different radicals:
  • For square roots, find a number that, when multiplied by itself, equals the original number.
  • For cube roots, find a number that, when cubed, equals the original number.
  • For fourth roots, find a number that, when raised to the fourth power, equals the original number.
  • For fifth roots, find a number that, when raised to the fifth power, equals the original number.
Example simplifications include \(\root[3]{27} = 3\), \(\root[4]{16} = 2\), and \(\root[5]{243} = 3\), each solving the radical into a simpler form.

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Most popular questions from this chapter

Simplify. Assume all variables are positive (a) \(\frac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}$$\left(\frac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}\)

Use the Quotient Property to simplify square roots. \(\sqrt{\frac{150 r^{3}}{256}}\)

Write with a rational exponent. (a) \(\sqrt[8]{r}\) (b) \(\sqrt[10]{s}\) (c) \(\sqrt[4]{t}\)

Write with a rational exponent. \(\sqrt[5]{u^{2}}\)(b) \((\sqrt[3]{6 x})^{5}\)(c)\(\sqrt[4]{\left(\frac{18 a}{5 b}\right)^{7}}\)

Simplify using absolute value signs as needed. (a) \(\sqrt{192 q^{3} r^{7}}\) (b) \(\sqrt[3]{54 m^{9} n^{10}}\)(c) \(\sqrt[4]{81 a^{9} b^{8}}\)
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