[tex]\left[\frac{8}{125}\right]^{\frac{1}{3}}[/tex]
Answer (1)
Rewrite the fraction 125 8 as 5 3 2 3 .
Apply the power of a quotient rule: [ 5 3 2 3 ] 3 1 = ( 5 3 ) 3 1 ( 2 3 ) 3 1 .
Apply the power rule: ( 5 3 ) 3 1 ( 2 3 ) 3 1 = 5 2 .
Simplify the expression to obtain the final result: 5 2 .
Explanation
Understanding the problem We are asked to evaluate the expression [ 125 8 ] 3 1 . This involves finding the cube root of a fraction.
Rewriting the fraction We can rewrite the fraction 125 8 as 5 3 2 3 , since 8 = 2 × 2 × 2 = 2 3 and 125 = 5 × 5 × 5 = 5 3 . Therefore, our expression becomes [ 5 3 2 3 ] 3 1 .
Applying the power of a quotient rule Using the property that ( b a ) n = b n a n , we can rewrite the expression as ( 5 3 ) 3 1 ( 2 3 ) 3 1 .
Applying the power rule Now, we use the property that ( a m ) n = a mn . So, ( 2 3 ) 3 1 = 2 3 × 3 1 = 2 1 = 2 and ( 5 3 ) 3 1 = 5 3 × 3 1 = 5 1 = 5 .
Simplifying the expression Therefore, the expression simplifies to 5 2 , which is equal to 0.4.
Final Answer Thus, [ 125 8 ] 3 1 = 5 2 = 0.4 .
Examples
Imagine you are designing a set of cubic building blocks. You want the volume of a larger block to be 8/125 cubic meters. To find the length of each side of the cube, you need to calculate the cube root of the volume, which is ( 125 8 ) 3 1 . This calculation tells you that each side of the cube should be 2/5 meters long. Understanding fractional exponents helps in scaling and designing objects proportionally.
