Evaluate the following.
Click on "Not a real number" if applicable.
(a) $-\sqrt[3]{125}= \square$
Not a real number
(b) $\sqrt[4]{-256}= \square$
Answer (2)
Calculate the cube root of 125: 3 125 = 5 .
Apply the negative sign: − 3 125 = − 5 .
Determine if 4 − 256 is a real number. Since the index is even and the radicand is negative, it is not a real number.
Final answers: (a) − 5 and (b) Not a real number.
Explanation
Understanding the Problem We are asked to evaluate two expressions involving roots. The first expression is − 3 125 , which involves a cube root. The second expression is 4 − 256 , which involves a fourth root.
Evaluating the Cube Root For part (a), we need to find the cube root of 125. The cube root of a number x is a value that, when multiplied by itself three times, equals x . In this case, we are looking for a number that, when cubed, equals 125. We know that 5 × 5 × 5 = 125 , so 3 125 = 5 . Since there is a negative sign in front of the cube root, the final answer for part (a) is − 5 .
Evaluating the Fourth Root For part (b), we need to find the fourth root of -256. The fourth root of a number x is a value that, when multiplied by itself four times, equals x . In this case, we are looking for a number that, when raised to the fourth power, equals -256. However, any real number raised to an even power (such as 4) will always be non-negative. Therefore, there is no real number that, when raised to the fourth power, equals -256. Thus, the fourth root of -256 is not a real number.
Final Answer Therefore, the answer to part (a) is -5, and the answer to part (b) is 'Not a real number'.
Examples
Understanding roots and radicals is essential in various fields, such as engineering and physics. For instance, when calculating the period of a pendulum, you use the square root of the length of the pendulum. Similarly, when dealing with volumes and scaling, cube roots are often used. Knowing when a root results in a real number or not is crucial in these calculations to ensure meaningful results.
For part (a), the answer is -5, as the cube root of 125 is 5. For part (b), the answer is 'Not a real number', since the fourth root of a negative number is not defined in real numbers.
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