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Chapter 7: Problem 17

Evaluate expression. \((-125)^{1 / 3}\)

Short Answer

Expert verified
-5

Step by step solution

01

Understand the Cube Root

The expression \((-125)^{1/3}\) represents the cube root of -125. The cube root of a number \(a\) is a value \(b\) such that \(b^3 = a\).
02

Find the Cube Root

To find the cube root of -125, consider a number that, when multiplied by itself three times, gives -125. Since \((-5) \times (-5) \times (-5) = -125\), the cube root of -125 is -5.
03

Express the Result

Thus, \((-125)^{1/3} = -5\).

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Key Concepts

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

Evaluating Expressions
Evaluating expressions involves calculating the value of an expression by substituting variables with numbers or computing the values of numeric expressions. For example, evaluating the expression \((-125)^{1 / 3}\) means finding the cube root of -125. To perform this calculation, understand the mathematical operation involved, replace any variables (if present) with given numbers, and follow the arithmetic or algebraic rules governing the operations. The final result is the value of the expression. Evaluating expressions is a key skill in solving math problems and real-life applications.
Cube Roots
Cube roots pertain to the operation of finding a number which, when multiplied by itself three times (cubed), yields the given number. In the case of the cube root of -125, we look for a number that satisfies \(b^3 = -125\). Considering different possibilities, we identify that \((-5) \times (-5) \times (-5) = -125\). Therefore, the cube root of -125 is -5. Cube roots can be evaluated by recognizing patterns, consulting cube root tables, or using a calculator.
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The radical symbol (鈭�) represents these roots, and expressions like \(\sqrt[3]{-125}\) or \(-125)^{1/3}\) symbolize the cube root of -125. Solving radical expressions usually requires simplification by finding the root value that, when raised to a specific power, matches the original number. For example, in evaluating \((-125)^{1 / 3}\), understanding the radical expression, simplifying under the radical sign, and knowing root properties are essential. These steps enable easy calculation and understanding of how radical expressions function in more complex math problems.

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Most popular questions from this chapter

Discussion Which one of the following expressions is not equivalent to the others? a) \((\sqrt[3]{x})^{4}\) b) \(\sqrt[4]{x^{3}}\) c) \(\sqrt[3]{x^{4}}\) d) \(x^{4 / 3}\) e) \(\left(x^{1 / 3}\right)^{4}\)

Solve each equation. $$(r-1)^{-2 / 3}=1$$

According to the new International America's Cup Class Rules, the maximum sail area in square meters for a yacht in the America's Cup race is given by the function $$S=\left(13.0368+7.84 D^{1 / 3}-0.8 L\right)^{2}$$ where \(D\) is the displacement in cubic meters \(\left(\mathrm{m}^{3}\right),\) and \(L\) is the length in meters (m). (www.sailing.com). Find the maximum sail area for a boat that has a displacement of \(18.42 \mathrm{m}^{3}\) and a length of \(21.45 \mathrm{m}\).

Simplify. $$\frac{\sqrt{5}}{3-\sqrt{y}}-\frac{\sqrt{5 y}}{3+\sqrt{y}}$$

Solve each equation. $$\sqrt[3]{2 w+3}=\sqrt[3]{w-2}$$
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