Evaluate the following.
Click on "Not a real number" if applicable.
(a) $-\sqrt[4]{625}=$ $\square$
(b) $\sqrt[3]{-27}=$ $\square$
Answer (2)
Calculate the fourth root of 625: 4 625 = 5 .
Apply the negative sign: − 4 625 = − 5 .
Calculate the cube root of -27: 3 − 27 = − 3 .
The final answers are -5 and -3: − 5 and − 3 .
Explanation
Problem Analysis We need to evaluate two expressions involving roots: (a) − 4 625 (b) 3 − 27
Evaluating (a) (a) To evaluate − 4 625 , we first find the fourth root of 625. We are looking for a number x such that x 4 = 625 . Since 5 4 = 625 , we have 4 625 = 5 . Then, we apply the negative sign to get − 4 625 = − 5 .
Evaluating (b) (b) To evaluate 3 − 27 , we are looking for a number y such that y 3 = − 27 . Since ( − 3 ) 3 = − 27 , we have 3 − 27 = − 3 .
Final Answer Therefore, (a) − 4 625 = − 5 (b) 3 − 27 = − 3
Examples
Understanding roots and exponents is crucial in many fields, such as physics and engineering. For instance, when calculating the period of a pendulum, you use square roots. If you're designing a bridge, understanding cube roots can help you calculate the volume of materials needed. These concepts are also fundamental in financial mathematics, such as calculating compound interest or depreciation.
After evaluating the expressions, we find that (a) − 4 625 = − 5 and (b) 3 − 27 = − 3 .
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