Sure! Let's approach the problem step-by-step by simplifying and combining the given exponents.
### Step 1: Simplify the First Expression
We start with the first expression:
[tex]\[
\left(x^{\frac{2}{9}}\right)^{\frac{3}{8}}
\][/tex]
Using the exponent power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we multiply the exponents:
[tex]\[
\left(x^{\frac{2}{9}}\right)^{\frac{3}{8}} = x^{\left(\frac{2}{9} \cdot \frac{3}{8}\right)}
\][/tex]
Calculate the exponent:
[tex]\[
\frac{2}{9} \cdot \frac{3}{8} = \frac{2 \times 3}{9 \times 8} = \frac{6}{72} = \frac{1}{12}
\][/tex]
So,
[tex]\[
\left(x^{\frac{2}{9}}\right)^{\frac{3}{8}} = x^{\frac{1}{12}}
\][/tex]
### Step 2: Combine All Expressions
Now, let's combine all the given expressions. We know from the simplified form of the first expression that:
[tex]\[
x^{\frac{1}{12}}
\][/tex]
The given expressions are:
[tex]\[
x^{\frac{5}{17}}, \quad x^{\frac{1}{12}}, \quad x^{\frac{11}{72}}, \quad x^{\frac{43}{72}}
\][/tex]
We need to combine these by adding their exponents.
### Step 3: Sum the Exponents
Let's list out all the exponents:
[tex]\[
\frac{1}{12}, \quad \frac{5}{17}, \quad \frac{1}{12}, \quad \frac{11}{72}, \quad \frac{43}{72}
\][/tex]
Sum these exponents:
[tex]\[
\frac{1}{12} + \frac{5}{17} + \frac{1}{12} + \frac{11}{72} + \frac{43}{72}
\][/tex]
Using the numerical results directly:
[tex]\[
\frac{1}{12} \approx 0.083333, \quad \frac{5}{17} \approx 0.294118, \quad \frac{1}{12} \approx 0.083333, \quad \frac{11}{72} \approx 0.152778, \quad \frac{43}{72} \approx 0.597222
\][/tex]
Add these values:
[tex]\[
0.083333 + 0.294118 + 0.083333 + 0.152778 + 0.597222 = 1.210784
\][/tex]
Finally, since these are all exponents of [tex]\(x\)[/tex], we combine them under a single exponent:
[tex]\[
x^{1.210784}
\][/tex]
Thus, after combining all the given expressions, we get:
[tex]\[
x^{1.210784}
\][/tex]
