Use the properties of exponents to solve for each variable.
[tex]\begin{array}{l}
4^8 \cdot 4^2=4^a \\
\left(2^4\right)^5=2^b \\
\frac{5^6}{5^2}=5^c
\end{array}[/tex]
a =
b =
c =
Answer (1)
Use the property x m c d o t x n = x m + n to find a : 4 8 c d o t 4 2 = 4 8 + 2 = 4 10 , so a = 10 .
Use the property ( x m ) n = x m c d o t n to find b : ( 2 4 ) 5 = 2 4 c d o t 5 = 2 20 , so b = 20 .
Use the property x n x m = x m − n to find c : 5 2 5 6 = 5 6 − 2 = 5 4 , so c = 4 .
The solutions are: a = 10 , b = 20 , c = 4
Explanation
Understanding the Problem We are given three equations involving exponents and asked to solve for the variables a, b, and c. We will use the properties of exponents to simplify each equation and find the values of the variables.
Solving for a The first equation is 4 8 c d o t 4 2 = 4 a . Using the property of exponents that x m c d o t x n = x m + n , we have 4 8 + 2 = 4 10 = 4 a . Therefore, a = 10 .
Solving for b The second equation is ( 2 4 ) 5 = 2 b . Using the property of exponents that ( x m ) n = x m c d o t n , we have 2 4 c d o t 5 = 2 20 = 2 b . Therefore, b = 20 .
Solving for c The third equation is 5 2 5 6 = 5 c . Using the property of exponents that x n x m = x m − n , we have 5 6 − 2 = 5 4 = 5 c . Therefore, c = 4 .
Final Answer Therefore, the values of the variables are a = 10 , b = 20 , and c = 4 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with data storage sizes (kilobytes, megabytes, gigabytes, etc.) or in finance when calculating compound interest. For example, if you invest 100 a t anann u a l in t eres t r a t eo f 5 A = 100(1.05)^t$. Exponents help us understand how investments grow over time.
