Simplify $\left(\frac{-1}{w^{10}}\right)^3$.
A. $\frac{3}{w^{13}}$
B. $\frac{-3}{w^{30}}$
C. $-\frac{1}{w^{30}}$
D. $-\frac{1}{w^{13}}$
Answer (1)
Apply the power of a quotient rule: ( w 10 − 1 ) 3 = ( w 10 ) 3 ( − 1 ) 3 .
Calculate ( − 1 ) 3 = − 1 .
Calculate ( w 10 ) 3 = w 30 using the power of a power rule.
Simplify the expression to get the final answer: − w 30 1 .
Explanation
Understanding the problem We are asked to simplify the expression ( w 10 − 1 ) 3 . This involves raising a fraction to a power. We will use the properties of exponents to simplify this expression.
Applying the power of a quotient rule First, we apply the power of a quotient rule, which states that ( b a ) n = b n a n . In our case, a = − 1 , b = w 10 , and n = 3 . So we have ( w 10 − 1 ) 3 = ( w 10 ) 3 ( − 1 ) 3 .
Calculating (-1)^3 Next, we calculate ( − 1 ) 3 . Since ( − 1 ) 3 = ( − 1 ) × ( − 1 ) × ( − 1 ) = − 1 , we have ( w 10 ) 3 ( − 1 ) 3 = ( w 10 ) 3 − 1 .
Simplifying the denominator Now, we simplify the denominator ( w 10 ) 3 . We use the power of a power rule, which states that ( a m ) n = a mn . In our case, a = w , m = 10 , and n = 3 . So we have ( w 10 ) 3 = w 10 × 3 = w 30 .
Final simplification Finally, we substitute this back into our expression: ( w 10 ) 3 − 1 = w 30 − 1 = − w 30 1 .
Examples
Imagine you are working with computer file sizes. If a file's size is represented as a fraction involving exponents, simplifying that fraction helps you understand the actual storage space it occupies. For example, if a file size is given by ( 2 10 1 ) 3 gigabytes, simplifying this expression to 2 30 1 gigabytes makes it easier to compare with other file sizes and storage capacities. This is also applicable in network bandwidth calculations, where data transfer rates are often expressed using exponents.
