Evaluate the following.
Click on "Not a real number" if applicable.
(a) $\sqrt[3]{-27}=$
Not a real number
(b) $\sqrt[4]{-81}=$
Answer (2)
The cube root of -27 is found by identifying a number that, when cubed, equals -27.
Since ( − 3 ) 3 = − 27 , then 3 − 27 = − 3 .
The fourth root of -81 is not a real number because any real number raised to an even power is non-negative.
Therefore, the answers are − 3 and Not a real number.
Explanation
Problem Analysis We are asked to evaluate two expressions: 3 − 27 and 4 − 81 . We need to determine if the results are real numbers.
Evaluating the Cube Root (a) To evaluate 3 − 27 , we need to find a number that, when raised to the power of 3, equals -27. Since ( − 3 ) 3 = ( − 3 ) × ( − 3 ) × ( − 3 ) = − 27 , we have 3 − 27 = − 3 .
Evaluating the Fourth Root (b) To evaluate 4 − 81 , we need to find a number that, when raised to the power of 4, equals -81. However, any real number raised to an even power (such as 4) will result in a non-negative number. Therefore, there is no real number that, when raised to the power of 4, equals -81. Thus, 4 − 81 is not a real number.
Final Answer Therefore, the final answers are: (a) 3 − 27 = − 3 (b) 4 − 81 is not a real number.
Examples
Understanding roots and radicals is crucial in various fields, such as physics and engineering. For example, when calculating the period of a pendulum, you use the square root of the length of the pendulum. Similarly, in electrical engineering, you might encounter cube roots when dealing with power calculations. Knowing how to evaluate roots, including those of negative numbers, allows for accurate and meaningful calculations in these real-world applications.
The cube root of -27 is -3, as ( − 3 ) 3 = − 27 . The fourth root of -81 is not a real number because no real number raised to an even power can yield a negative result. Therefore, the answers are -3 and not a real number, respectively.
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