Certainly! Let's solve the equation [tex]\( 8^{3^{x+1}} = 2^{a^{x+1}} \)[/tex] and find [tex]\( a \)[/tex] step by step.
### Step 1: Express 8 in terms of a power of 2
We know that [tex]\( 8 \)[/tex] can be written as [tex]\( 2^3 \)[/tex] because [tex]\( 2^3 = 8 \)[/tex].
So the left side of the equation [tex]\( 8^{3^{x+1}} \)[/tex] can be rewritten as:
[tex]\[ 8^{3^{x+1}} = \left(2^3\right)^{3^{x+1}}. \][/tex]
### Step 2: Simplify the left side using properties of exponents
Using the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ \left(2^3\right)^{3^{x+1}} = 2^{3 \cdot 3^{x+1}} = 2^{3^{x+1+1}} = 2^{3 \cdot 3^{x + 1}}. \][/tex]
Thus, the equation now becomes:
[tex]\[ 2^{3 \cdot 3^{x+1}} = 2^{a^{x+1}}. \][/tex]
### Step 3: Equate the exponents
Since the bases on both sides of the equation are the same, we can equate the exponents:
[tex]\[ 3 \cdot 3^{x+1} = a^{x+1}. \][/tex]
### Step 4: Solve for [tex]\( a \)[/tex]
We need to isolate [tex]\( a \)[/tex] in the equation [tex]\( 3 \cdot 3^{x+1} = a^{x+1} \)[/tex]. Notice the left side can be expressed as a single exponent:
[tex]\[ 3 \cdot 3^{x+1} = 3^{x+1+1} = 3^{x+2}. \][/tex]
Therefore, the equation becomes:
[tex]\[ 3^{x+2} = a^{x+1}. \][/tex]
To isolate [tex]\( a \)[/tex], we need to take the [tex]\((x+1)\)[/tex]-th root on both sides:
[tex]\[ a = (3^{x+2})^{\frac{1}{x+1}}. \][/tex]
### Final Result
Thus, the solution for [tex]\( a \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[ a = (3^{x+2})^{\frac{1}{x+1}}. \][/tex]
