Solve:
$\frac{512^{x-2}}{\left(\frac{1}{64}\right)^{3 x}}=512$
A. $x=-1$
B. $x=0$
C. $x=1$
D. no solution
Answer (1)
Rewrite the equation using powers of 2: ( 2 − 6 ) 3 x ( 2 9 ) x − 2 = 2 9 .
Simplify the exponents: 2 − 18 x 2 9 ( x − 2 ) = 2 9 .
Combine the terms: 2 9 x − 18 + 18 x = 2 9 , which simplifies to 2 27 x − 18 = 2 9 .
Equate the exponents and solve for x : 27 x − 18 = 9 , so x = 1 .
1
Explanation
Understanding the Problem We are given the equation ( 64 1 ) 3 x 51 2 x − 2 = 512 and asked to solve for x . We are given possible answers x = − 1 , x = 0 , x = 1 , or no solution. We will rewrite the equation using powers of 2.
Rewriting with Powers of 2 First, we rewrite the numbers in the equation as powers of 2. We have 512 = 2 9 and 64 1 = 2 6 1 = 2 − 6 . Substituting these into the equation gives ( 2 − 6 ) 3 x ( 2 9 ) x − 2 = 2 9 .
Simplifying Exponents Next, we simplify the exponents. Using the property ( a b ) c = a b c , we have 2 − 18 x 2 9 ( x − 2 ) = 2 9 .
Combining Terms Now, we use the property a c a b = a b − c to combine the terms on the left side: 2 9 ( x − 2 ) − ( − 18 x ) = 2 9 .
Simplifying the Exponent We simplify the exponent: 2 9 x − 18 + 18 x = 2 9 , which simplifies to 2 27 x − 18 = 2 9 .
Equating Exponents Since the bases are equal, the exponents must be equal: 27 x − 18 = 9.
Solving for x Now we solve for x :
27 x = 9 + 18 = 27 , so x = 27 27 = 1.
Checking the Solution Finally, we check if x = 1 is a valid solution by substituting it back into the original equation: ( 64 1 ) 3 ( 1 ) 51 2 1 − 2 = ( 64 1 ) 3 51 2 − 1 = 6 4 3 1 512 1 = 512 1 ⋅ 6 4 3 = 512 6 4 3 = 2 9 ( 2 6 ) 3 = 2 9 2 18 = 2 9 = 512. Thus, x = 1 is indeed a solution.
Examples
When dealing with exponential growth or decay, such as in population studies or radioactive decay, solving equations with exponents is crucial. For instance, if you're modeling the population growth of a bacteria colony, you might use an equation similar to the one above to determine how long it takes for the colony to reach a certain size. By understanding how to manipulate and solve exponential equations, you can make accurate predictions and informed decisions in various scientific and real-world scenarios.
