Let's solve the given equation step by step:
[tex]\[
\sqrt[x]{2^{x+2}} \cdot \sqrt[3]{64} = 2
\][/tex]
### Step 1: Rewrite the equation with simpler exponential terms
First, let's break down each term separately.
1. Express [tex]\(\sqrt[x]{2^{x+2}}\)[/tex] in terms of exponents:
[tex]\[
\sqrt[x]{2^{x+2}} = (2^{x+2})^{1/x} = 2^{(x+2)/x} = 2^{1+2/x}
\][/tex]
2. Simplify [tex]\(\sqrt[3]{64}\)[/tex]:
[tex]\[
\sqrt[3]{64} = 64^{1/3} = (2^6)^{1/3} = 2^{6/3} = 2^2 = 4
\][/tex]
### Step 2: Substitute the simplified terms back into the equation
The initial equation simplifies to:
[tex]\[
2^{1+2/x} \cdot 4 = 2
\][/tex]
Since [tex]\(4 = 2^2\)[/tex], we can rewrite the equation as:
[tex]\[
2^{1+2/x} \cdot 2^2 = 2
\][/tex]
### Step 3: Combine the exponents
Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
2^{1+2/x + 2} = 2
\][/tex]
Combine the exponents:
[tex]\[
2^{3 + 2/x} = 2^1
\][/tex]
### Step 4: Equate the exponents
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
3 + 2/x = 1
\][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Subtract 3 from both sides:
[tex]\[
\frac{2}{x} = 1 - 3
\][/tex]
[tex]\[
\frac{2}{x} = -2
\][/tex]
Multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
2 = -2x
\][/tex]
Divide both sides by -2:
[tex]\[
x = -1
\][/tex]
Thus, the solution to the equation is:
[tex]\[
\boxed{-1}
\][/tex]
