Answer :
To solve the equation [tex]\( 36^{12-m} = 6^{2m} \)[/tex], follow these steps:
1. Express 36 as a power of 6:
Since [tex]\( 36 = 6^2 \)[/tex], we can rewrite the equation as:
[tex]\[ (6^2)^{12-m} = 6^{2m} \][/tex]
2. Simplify the left-hand side:
Applying the power of a power property [tex]\((a^b)^c = a^{bc}\)[/tex]:
[tex]\[ 6^{2(12-m)} = 6^{2m} \][/tex]
3. Simplify the exponent:
Multiply the exponents on the left-hand side:
[tex]\[ 6^{24 - 2m} = 6^{2m} \][/tex]
4. Since the bases are the same, set the exponents equal:
We have the same base (6) on both sides, so we can set the exponents equal to one another:
[tex]\[ 24 - 2m = 2m \][/tex]
5. Solve for [tex]\( m \)[/tex]:
Combine like terms:
[tex]\[ 24 = 4m \][/tex]
Isolate [tex]\( m \)[/tex] by dividing both sides by 4:
[tex]\[ m = \frac{24}{4} \][/tex]
[tex]\[ m = 6 \][/tex]
Therefore, the value of [tex]\( m \)[/tex] is [tex]\( \boxed{6} \)[/tex].
1. Express 36 as a power of 6:
Since [tex]\( 36 = 6^2 \)[/tex], we can rewrite the equation as:
[tex]\[ (6^2)^{12-m} = 6^{2m} \][/tex]
2. Simplify the left-hand side:
Applying the power of a power property [tex]\((a^b)^c = a^{bc}\)[/tex]:
[tex]\[ 6^{2(12-m)} = 6^{2m} \][/tex]
3. Simplify the exponent:
Multiply the exponents on the left-hand side:
[tex]\[ 6^{24 - 2m} = 6^{2m} \][/tex]
4. Since the bases are the same, set the exponents equal:
We have the same base (6) on both sides, so we can set the exponents equal to one another:
[tex]\[ 24 - 2m = 2m \][/tex]
5. Solve for [tex]\( m \)[/tex]:
Combine like terms:
[tex]\[ 24 = 4m \][/tex]
Isolate [tex]\( m \)[/tex] by dividing both sides by 4:
[tex]\[ m = \frac{24}{4} \][/tex]
[tex]\[ m = 6 \][/tex]
Therefore, the value of [tex]\( m \)[/tex] is [tex]\( \boxed{6} \)[/tex].
