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Chapter 8: Problem 13

Simplify. (a) \(\sqrt[3]{512}\) (b) \(\sqrt[4]{81}\) (c) \(\sqrt[5]{1}\)

Short Answer

Expert verified
a) 8 b) 3 c) 1

Step by step solution

01

Simplifying \(\root[3]{512} \)

Identify the number to find the cube root of: 512. The cube root of a number is the value that, when multiplied by itself three times, gives the original number. In this case, \(8^3=512 \), so \(\root[3]{512} \) simplifies to 8.
02

Simplifying \(\root[4]{81} \)

Identify the number to find the fourth root of: 81. The fourth root of a number is the value that, when multiplied by itself four times, gives the original number. In this case, \(3^4=81 \), so \(\root[4]{81} \) simplifies to 3.
03

Simplifying \(\root[5]{1} \)

Identify the number to find the fifth root of: 1. The fifth root of a number is the value that, when multiplied by itself five times, gives the original number. In this case, \(1^5=1 \), so \(\root[5]{1} \) simplifies to 1.

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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 of a number is the value that, when multiplied by itself three times, equals the original number. Remember that the cube root is denoted using the radical sign with a small three above it, like this: \(\root[3]{a} \).

Here's a simple way to understand cube roots:
  • Start with your number, for example, 512.
  • Find a number that, when multiplied by itself three times (or cubed), equals your starting number.
In this case, 8 multiplied by itself three times is 8 * 8 * 8 = 512.
Therefore, the cube root of 512 is 8. This can be written as \(\root[3]{512} = 8 \).
Understanding cube roots is fundamental for solving higher-level algebraic equations and can even help in various real-life applications like physics and engineering.
fourth root
The fourth root of a number is the value that, when multiplied by itself four times, equals the original number. It is represented with a radical sign and a small four above it, like this: \(\root[4]{a} \).

Let's break down how to find a fourth root:
  • Start with your number, for example, 81.
  • Look for a number that, when you multiply it by itself four times, gives you the starting number.
For 81, the number we need is 3, because 3 * 3 * 3 * 3 = 81.
Therefore, the fourth root of 81 is 3, which can be written as \(\root[4]{81} = 3 \).
Knowing how to calculate fourth roots can be particularly useful in fields like statistics and computer science where certain types of data transformations are necessary.
fifth root
The fifth root of a number is the value that, when multiplied by itself five times, equals the original number. This root is represented by a radical sign with a small five above it, like this: \(\root[5]{a} \).

Discovering how to find a fifth root can be straightforward:
  • Start with your number, in this case, 1.
  • Determine which number, when multiplied by itself five times, results in the original number.
Here, 1 is quite easy since 1 * 1 * 1 * 1 * 1 = 1.
Thus, the fifth root of 1 is simply 1, which can be expressed as \(\root[5]{1} = 1 \).
Understanding fifth roots is essential in more advanced areas of mathematics, as well as in applications like finance and engineering where exponential growth patterns are analyzed.

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Most popular questions from this chapter

Use the Product Property to simplify radical expressions. . \(\sqrt{96}\)

Use the Quotient Property to simplify square roots. (a) \(\sqrt{\frac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}\) \\}(b) \(\sqrt[3]{\frac{24 m^{9} n^{7}}{375 m^{4} n}}\)(c) \(\sqrt[4]{\frac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}\)

Simplify. (a) \((-1000)^{\frac{1}{3}}\) (b) \(-1000^{\frac{1}{3}}\) (c)\((1000)^{-\frac{1}{3}}\)

Simplify. (a) \(-64^{\frac{3}{2}}\)(b) \(-64^{-\frac{3}{2}}\) (c)\((-64)^{\frac{3}{2}}\)

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