Given that
[tex]8^{2 x+3} \times \frac{1}{4^{3 x}}=2^{x+3}[/tex]
what is the value of [tex]x[/tex] ?
A. 0
B. 3
C. 6
D. [tex]\frac{1}{2}[/tex]
E. [tex]\frac{3}{2}[/tex]
Answer (1)
Express all terms as powers of 2: 8 2 x + 3 × 4 3 x 1 = ( 2 3 ) 2 x + 3 × ( 2 2 ) 3 x 1 .
Simplify the equation: 2 6 x + 9 × 2 − 6 x = 2 x + 3 .
Combine the exponents: 2 9 = 2 x + 3 .
Equate the exponents and solve for x : 9 = x + 3 , so x = 6 .
Explanation
Understanding the Problem We are given the equation ".\times \frac{1}{4^{3x}} = 2^{x+3}"> 8 2 x + 3 ". > ". × 4 3 x 1 = 2 x + 3 and we need to find the value of x .
Expressing as Powers of 2 First, we express all terms in the equation as powers of 2. We know that 8 = 2 3 and 4 = 2 2 . So, we can rewrite the equation as ( 2 3 ) 2 x + 3 × ( 2 2 ) 3 x 1 = 2 x + 3 .
Simplifying the Equation Next, we simplify the equation using the properties of exponents: ( 2 3 ) 2 x + 3 = 2 3 ( 2 x + 3 ) = 2 6 x + 9 and ( 2 2 ) 3 x = 2 6 x . So, the equation becomes 2 6 x + 9 × 2 6 x 1 = 2 x + 3 .
Rewriting the Equation We can rewrite 2 6 x 1 as 2 − 6 x . Thus, the equation becomes 2 6 x + 9 × 2 − 6 x = 2 x + 3 .
Combining Exponents Now, we simplify the left side of the equation by combining the exponents: 2 ( 6 x + 9 ) + ( − 6 x ) = 2 x + 3 . This simplifies to 2 9 = 2 x + 3 .
Equating Exponents Since the bases are equal, we can equate the exponents: 9 = x + 3 .
Solving for x Finally, we solve for x : x = 9 − 3 = 6 . Therefore, the value of x is 6.
Examples
Understanding exponential equations is crucial in various fields, such as calculating compound interest in finance or modeling population growth in biology. For instance, if a population doubles every year, the equation \times". 2^t"> P = P 0 ". > × ". 2 t describes the population P after t years, where P 0 is the initial population. Solving such equations helps predict future population sizes or determine the time it takes for a population to reach a certain level. This concept is also fundamental in understanding radioactive decay and carbon dating in archaeology.
