Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Roots Evaluate each radical by calculator, retaining the proper number of digits in your answer: $$\sqrt[3]{-2.774}$$

Short answer

The cube root of -2.774 to four significant digits is approximately -1.407.

Step-by-step solution

Identify the Type of Root and the Value Inside

Determine the kind of root you are dealing with and the value that needs to be calculated. Here, it is a cube root, denoted as \(\sqrt[3]{-2.774}\).

Consider the Sign of the Value Inside the Radical

Recognize that cube roots can be taken of negative numbers (unlike even roots like square roots), and in this case, since -2.774 is negative, the result will also be negative.

Use a Calculator to Find the Cube Root

Enter the value 2.774 into a calculator and take the cube root. Remember to apply the correct sign (negative) to the result.

Retain Proper Significant Digits

The input has four significant digits, so the output should also retain four significant digits. Round the result, if necessary, to match the number of significant digits in the original number.

Write the Final Answer

After rounding, if necessary, to four significant digits, write down the final answer with the correct sign.

Key concepts

Radical Expressions

Radical expressions, such as square roots, cube roots, and higher order roots, are mathematical operations that involve finding a number, which when raised to a certain power, gives the original value under the radical sign. When dealing with radical expressions, the type of root is crucial: square roots are denoted by \( \sqrt{} \), cube roots by \( \sqrt[3]{} \), and so on.

In simplifying radical expressions, we look for the prime factorization of the number inside the radical and then simplify accordingly. It is important to remember that the radical itself signifies a specific mathematical operation and retains its own properties and rules—such as the fact that you cannot take the even root of a negative number without delving into complex numbers.

Cube Root Calculation

Calculating a cube root, represented by \( \sqrt[3]{} \) is the process of finding a number that, when multiplied by itself three times (raised to the third power), yields the original number within the radical sign. Unlike square roots, cube roots can be taken of negative numbers since a negative number multiplied by itself three times will result in a negative product.

When using a calculator to compute the cube root, simply enter the value—whether positive or negative—and apply the cube root function. In a mathematical proof or solution where calculators aren't allowed, you would need to rely on estimation or numerical methods to find cube roots. However, adults usually have access to calculators, thus cube roots of non-perfect cubes are typically solved using such electronic devices.

Negative Roots

The concept of negative roots is intertwined with the type of root being taken. For \( \sqrt[n]{} \), where \( n \) is an even number, negative roots cannot result in real numbers since the even power of any real number (positive or negative) will always be positive. On the other hand, for odd roots like cube roots, it is perfectly viable to have a negative number since an odd number of negative factors will result in a negative product.

It is essential to pay attention to the sign of the number inside the root to avoid errors in calculation. When finding a negative root, particularly with cube roots, the result carries the negative sign, as illustrated in the textbook example with the cube root of \( -2.774 \). Ensuring that the resulting value has the correct sign is a critical step in properly solving radical expressions that contain negative values.