To solve the equation [tex]\( 2^{2m - 3} = 8^m \)[/tex], we proceed with the following steps:
1. Rewrite 8 as a power of 2:
We know that [tex]\( 8 \)[/tex] can be expressed as [tex]\( 2^3 \)[/tex]. Therefore, we can rewrite the original equation by substituting [tex]\( 8 \)[/tex] with [tex]\( 2^3 \)[/tex]:
[tex]\[
2^{2m - 3} = (2^3)^m
\][/tex]
2. Simplify the right-hand side:
When we have an exponent raised to another exponent, we multiply the exponents. Thus, [tex]\( (2^3)^m \)[/tex] becomes [tex]\( 2^{3m} \)[/tex]:
[tex]\[
2^{2m - 3} = 2^{3m}
\][/tex]
3. Set the exponents equal:
Since the bases (2) are the same on both sides of the equation, we can set the exponents equal to each other:
[tex]\[
2m - 3 = 3m
\][/tex]
4. Solve for [tex]\( m \)[/tex]:
To find [tex]\( m \)[/tex], we solve the equation [tex]\( 2m - 3 = 3m \)[/tex]. First, we isolate [tex]\( m \)[/tex] by subtracting [tex]\( 2m \)[/tex] from both sides:
[tex]\[
2m - 3 - 2m = 3m - 2m
\][/tex]
This simplifies to:
[tex]\[
-3 = m
\][/tex]
Therefore, the solution to the equation [tex]\( 2^{2m - 3} = 8^m \)[/tex] is:
[tex]\[
m = -3
\][/tex]
