For what value of [tex]$n$[/tex] does [tex]$\frac{216^n}{\left(\frac{1}{36}\right)^{3 n}}=216$[/tex]?
Answer (1)
Rewrite the equation using exponent rules.
Express both 216 and 36 as powers of 6.
Simplify the equation and equate the exponents.
Solve for n : n = 3 1 .
The value of n is 3 1 .
Explanation
Understanding the Problem We are given the equation ( 36 1 ) 3 n 21 6 n = 216 and asked to find the value of n that satisfies the equation.
Rewriting the Equation We can rewrite the equation using properties of exponents. First, rewrite 36 1 as 3 6 − 1 . So the equation becomes ( 3 6 − 1 ) 3 n 21 6 n = 216 .
Simplifying the Denominator Next, simplify the denominator using the power of a power rule: ( a m ) n = a mn . Thus, ( 3 6 − 1 ) 3 n = 3 6 − 3 n . The equation is now 3 6 − 3 n 21 6 n = 216 .
Rewriting as a Product To simplify further, we can rewrite the fraction as a product by bringing the denominator to the numerator and changing the sign of the exponent: 21 6 n ⋅ 3 6 3 n = 216 .
Expressing as Powers of 6 Now, express both 216 and 36 as powers of 6. We know that 216 = 6 3 and 36 = 6 2 . Substituting these into the equation, we get ( 6 3 ) n ⋅ ( 6 2 ) 3 n = 6 3 .
Simplifying Exponents Using the power of a power rule again, we have 6 3 n ⋅ 6 6 n = 6 3 .
Combining Exponents When multiplying exponential expressions with the same base, we add the exponents: 6 3 n + 6 n = 6 3 , which simplifies to 6 9 n = 6 3 .
Equating Exponents Since the bases are equal, we can equate the exponents: 9 n = 3 .
Solving for n Finally, solve for n by dividing both sides by 9: n = 9 3 = 3 1 .
Final Answer Therefore, the value of n that satisfies the equation is 3 1 .
Examples
Understanding exponential equations is crucial in various fields, such as finance and computer science. For instance, calculating compound interest involves exponential growth. If you invest an amount of money that grows exponentially, solving equations like the one above can help you determine how long it takes for your investment to reach a certain value. Similarly, in computer science, exponential functions are used to analyze the complexity of algorithms, and solving exponential equations can help optimize performance.
