a) $\left[(2)^2\right]^{-1}$
b) $\left\{\left[\left(\frac{122}{15}\right)^5\right]^0\right\}^{-3}=$
Answer (1)
For expression a), apply the power of a power rule and the definition of a negative exponent to get [ ( 2 ) 2 ] − 1 = 2 − 2 = 4 1 .
For expression b), use the property that any non-zero number to the power of 0 is 1, and then simplify to get { [ ( 15 122 ) 5 ] 0 } − 3 = 1 − 3 = 1 .
The simplified form of expression a) is 4 1 .
The simplified form of expression b) is 1 .
Explanation
Introduction We are asked to simplify two expressions involving exponents. Let's tackle them one at a time.
Simplifying the first expression a) We have the expression [ ( 2 ) 2 ] − 1 . To simplify this, we use the rule that ( a m ) n = a m × n . In our case, a = 2 , m = 2 , and n = − 1 . Therefore, we have [ ( 2 ) 2 ] − 1 = 2 2 × ( − 1 ) = 2 − 2 Recall that a − n = a n 1 . Thus, 2 − 2 = 2 2 1 = 4 1 So, the simplified form of the expression is 4 1 .
Simplifying the second expression b) Now let's simplify the second expression: { [ ( 15 122 ) 5 ] 0 } − 3 . We start by simplifying the innermost expression. Recall that any non-zero number raised to the power of 0 is 1, i.e., a 0 = 1 for a = 0 . Therefore, ( 15 122 ) 5 = some non-zero number [ ( 15 122 ) 5 ] 0 = 1 Now we have { [ ( 15 122 ) 5 ] 0 } − 3 = { 1 } − 3 . Since 1 raised to any power is 1 , we have 1 − 3 = 1 3 1 = 1 1 = 1 So, the simplified form of the expression is 1 .
Final Answer Therefore, the simplified forms of the given expressions are: a) 4 1 b) 1
Examples
Understanding exponents is crucial in many fields, such as calculating compound interest, where the initial investment grows exponentially over time. For example, if you invest P dollars at an annual interest rate r compounded n times per year for t years, the future value A of the investment is given by the formula: A = P ( 1 + n r ) n t Simplifying expressions with exponents, as we did in this problem, is a fundamental skill needed to work with such formulas and understand the growth of investments or other exponential phenomena.
