$\frac{20^k}{20^4}=20^9$
$k=$
Answer (1)
Apply the exponent rule a n a m = a m − n to get 2 0 k − 4 = 2 0 9 .
Equate the exponents: k − 4 = 9 .
Solve for k by adding 4 to both sides: k = 9 + 4 .
Calculate the value of k : 13 .
Explanation
Understanding the Problem We are given the equation 2 0 4 2 0 k = 2 0 9 and we need to find the value of k .
Applying Exponent Rules To solve this equation, we will use the property of exponents that states a n a m = a m − n . Applying this property to our equation, we get 2 0 k − 4 = 2 0 9 .
Equating Exponents Since the bases are equal (both sides have a base of 20), we can equate the exponents. This gives us the equation k − 4 = 9 .
Solving for k Now, we solve for k by adding 4 to both sides of the equation: k = 9 + 4 .
Calculating k Performing the addition, we find that k = 13 .
Examples
Understanding exponential equations is crucial in various fields, such as calculating compound interest. For instance, if you invest money in an account that compounds interest annually, the equation A = P ( 1 + r ) t models the amount A you'll have after t years, where P is the principal and r is the interest rate. Solving for t or r often involves using logarithms, which are closely related to exponential functions. This concept helps in financial planning and understanding investment growth.
