Answer :
To solve the equation [tex]\( 3^{2x} + 12 = 39 \)[/tex], let's follow these steps:
1. Isolate the exponential term:
[tex]\[ 3^{2x} + 12 = 39 \][/tex]
Subtract 12 from both sides to isolate the exponential term:
[tex]\[ 3^{2x} = 39 - 12 \][/tex]
Simplify the right-hand side:
[tex]\[ 3^{2x} = 27 \][/tex]
2. Rewrite the right-hand side with the same base:
Notice that 27 can be written as a power of 3:
[tex]\[ 27 = 3^3 \][/tex]
Therefore, the equation becomes:
[tex]\[ 3^{2x} = 3^3 \][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are base 3), we can set the exponents equal:
[tex]\[ 2x = 3 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[ x = \frac{3}{2} \][/tex]
Thus, the solution to the equation [tex]\( 3^{2x} + 12 = 39 \)[/tex] is:
[tex]\[ x = \frac{3}{2} \][/tex]
The correct answer is:
[tex]\[ \boxed{\text{B. } x = \frac{3}{2}} \][/tex]
1. Isolate the exponential term:
[tex]\[ 3^{2x} + 12 = 39 \][/tex]
Subtract 12 from both sides to isolate the exponential term:
[tex]\[ 3^{2x} = 39 - 12 \][/tex]
Simplify the right-hand side:
[tex]\[ 3^{2x} = 27 \][/tex]
2. Rewrite the right-hand side with the same base:
Notice that 27 can be written as a power of 3:
[tex]\[ 27 = 3^3 \][/tex]
Therefore, the equation becomes:
[tex]\[ 3^{2x} = 3^3 \][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are base 3), we can set the exponents equal:
[tex]\[ 2x = 3 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[ x = \frac{3}{2} \][/tex]
Thus, the solution to the equation [tex]\( 3^{2x} + 12 = 39 \)[/tex] is:
[tex]\[ x = \frac{3}{2} \][/tex]
The correct answer is:
[tex]\[ \boxed{\text{B. } x = \frac{3}{2}} \][/tex]
