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Chapter 3: Problem 19

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[3]{27} $$

Short Answer

Expert verified
The value of \( \sqrt[3]{27} \) is 3.

Step by step solution

01

Identify the Expression

The given expression is \( \sqrt[3]{27} \), which is read as 'the cube root of 27'. Our task is to find the number that, when multiplied by itself three times (cubed), gives 27.
02

Find the Cube Root

To find the cube root, we look for a number that satisfies the equation \( x^3 = 27 \). By testing, we find that \( 3^3 = 3 \times 3 \times 3 = 27 \). Thus, \( 3 \) is the cube root of \( 27 \).
03

Confirm the Result

Verify the result by ensuring \( 3^3 = 27 \). Calculating gives \( 3 \times 3 = 9 \) and then \( 9 \times 3 = 27 \). Therefore, our computation is correct.

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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 fascinating mathematical concept, especially when dealing with real numbers. When we talk about the cube root, we are looking for a number that, when multiplied by itself twice more (cubed), results in the original number under the cube root. In simpler terms, for a number \( x \), the cube root is the value that fulfills the equation \( x^3 = \text{original number} \).
For example, if we consider \( \sqrt[3]{27} \):
  • We are searching for a number which, when used in the formula \( x^3 \), results in 27.
  • This number turns out to be 3, because \( 3 \times 3 \times 3 = 27 \).
Cube roots are useful because they help us find one of the three identical factors of a number when raised to the power of three. This is quite different from square roots, which only deal with two identical factors.
Evaluating Expressions
Evaluating expressions involves solving or simplifying a mathematical phrase to reach a singular value. This includes any form of mathematical notations, such as roots, exponents, and basic operations like addition and multiplication.
In the context of real numbers, evaluating an expression means finding a real number that represents the value of the expression.
Here are some general steps you can follow when evaluating expressions with cube roots:
  • Identify the operation needed to solve the expression. This involves recognizing symbols like exponents or roots.
  • Apply arithmetic operations accurately, while following the order of operations.
  • Always double-check your work by reversing the operation (e.g., if you find a cube root, cube it to verify).
By following these steps, you can accurately evaluate expressions containing cube roots and other complicated mathematical operations.
Exponents
Exponents are a shorthand way of expressing repeated multiplication. In mathematical terms, if a number \( a \) appears with an exponent \( n \), it means multiplying \( a \) by itself \( n \) times. This is written as \( a^n \).

The role of exponents is crucial when dealing with expressions that include powers, such as cube roots. When you see \( 3^3 \), this signifies:
  • Take the base number 3 and multiply it by itself two more times.
  • This can be expanded into \( 3 \times 3 \times 3 \), which equals 27.

Exponents help simplify expressions and enable you to succinctly represent larger calculations. Understanding them is essential not just for cube roots, but for many areas of math and real-world applications, such as in scientific notation or when dealing with growth calculations like compound interest.

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Most popular questions from this chapter

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ 3 \sqrt{2} \cdot \sqrt{10} $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 3-\sqrt{y}=1 $$

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 5+\sqrt[3]{a+2}=3 $$

The lengths of the legs of a right triangle in feet are \((3+\sqrt{3})\) and \((3-\sqrt{3})\) a. Find the length of the hypotenuse of the triangle. b. Express, in simplest form, the perimeter of the triangle. c. What is the area of the triangle?

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{5 x}-\sqrt{x+4}=3 $$
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