Final answer:
To evaluate (-7)¹/³ · ( 1/56 )^¹/³, calculate the cube root and the fifth power separately for each term, and then multiply the results. The final result, considering significant figures, is -525.
Step-by-step explanation:
To evaluate the expression (-7)¹/³ · ( 1/56 )^¹/³, we need to apply the rules of exponents for multiplication and dealing with fractional exponents. First, we can deal with each part of the expression separately.
For (-7)¹/³, this means we want to take the cube root of -7 and then raise it to the power of 5. The cube root of -7 is -7 because (-7)³ = -343. Raising -7 to the 5th power gives us -16807.
Now for (1/56)¹/³, we apply the same principle. The cube root of 1/56 is 1/√56. Raising this to the 5th power, we essentially raise 1 to the 5th power, which remains 1, and then raise √56 to the 5th power. Since 56 is 7×8, the cube root of 56 is the cube root of 7 times the cube root of 8, which is 2. Thus, the final answer is 1/(2¹), which is 1/32.
Multiplying -16807 by 1/32, we get -525.21875. However, when we consider significant figures, we need to round to the least number of decimal places in the given values, which in this case, none of the original numbers have decimal places. Therefore, we should round to a whole number, which gives us -525.