$\begin{array}{l}
\frac{7^2 \cdot 7^8}{7^4}=\frac{7^a}{7^4}=7^b \\
a=10
\end{array}$
COMPLETE
$\begin{array}{l}
\frac{2^b}{4}=\frac{2^b}{2^a}=2^b=c \\
a=\square, b=\square, c=\square
\end{array}$
Answer (2)
The values found are a = 10, b = 6, and c = 64 after applying simplification rules for exponents. First, we simplified the left side to determine b, and then used b to find the relationship in the second equation to solve for c. The final values reflect proper application of exponent rules.
;
Simplify the first equation using exponent rules to find the value of b .
Substitute the value of b into the second equation.
Use exponent rules to find the value of a .
Calculate the value of c using c = 2 b .
The final answers are: a = 2 , b = 6 , c = 64 , so a = 2 , b = 6 , c = 64 .
Explanation
Understanding the Problem We are given the equation 7 4 7 2 × 7 8 = 7 4 7 a = 7 b and a = 10 . We need to complete the equation 4 2 b = 2 a 2 b = 2 b = c by finding the values of a , b , and c .
Simplifying the Exponents First, let's simplify the expression 7 4 7 2 × 7 8 . Using the exponent rule x m × x n = x m + n , we have 7 2 × 7 8 = 7 2 + 8 = 7 10 . So, the expression becomes 7 4 7 10 .
Solving for b Now, using the exponent rule x n x m = x m − n , we have 7 4 7 10 = 7 10 − 4 = 7 6 . Thus, we have 7 4 7 a = 7 b which simplifies to 7 4 7 10 = 7 6 . This means 7 b = 7 6 , so b = 6 .
Substituting b = 6 Now we move on to the second equation: 4 2 b = 2 a 2 b = 2 b = c . We already found that b = 6 , so we can substitute this value into the equation: 4 2 6 = 2 a 2 6 = 2 6 = c .
Solving for a Since 4 = 2 2 , we can rewrite the equation as 2 2 2 6 = 2 a 2 6 = 2 6 = c . Now, using the exponent rule x n x m = x m − n , we have 2 2 2 6 = 2 6 − 2 = 2 4 . Therefore, 2 a 2 6 = 2 4 , which implies 2 a = 2 2 , so a = 2 .
Solving for c Finally, we find c = 2 b = 2 6 . Calculating 2 6 , we get 2 6 = 64 . Therefore, c = 64 .
Final Answer Thus, we have found the values of a , b , and c : a = 2 , b = 6 , and c = 64 .
Examples
Understanding exponents is crucial in many real-world applications, such as calculating compound interest. For instance, if you invest money in a bank account with compound interest, the amount of money you have after a certain period can be calculated using exponential growth. Similarly, in computer science, exponents are used to measure the complexity of algorithms. Knowing how to manipulate and simplify exponential expressions helps in optimizing code and understanding the efficiency of different algorithms.
