To solve the expression [tex]\(6 \sqrt[5]{3^5 x^{45}} \cdot \sqrt[8]{2^8 x}\)[/tex] and express it in the form [tex]\(a x^b\)[/tex], we can proceed with detailed steps as follows:
1. Simplifying each term inside the expression:
- The first part of the expression is [tex]\( \sqrt[5]{3^5 x^{45}} \)[/tex]:
[tex]\[
\sqrt[5]{3^5 x^{45}} = (3^5 \cdot x^{45})^{\frac{1}{5}} = 3^{\frac{5}{5}} \cdot x^{\frac{45}{5}} = 3^1 \cdot x^9 = 3 \cdot x^9
\][/tex]
- The second part of the expression is [tex]\( \sqrt[8]{2^8 x} \)[/tex]:
[tex]\[
\sqrt[8]{2^8 x} = (2^8 \cdot x)^{\frac{1}{8}} = 2^{\frac{8}{8}} \cdot x^{\frac{1}{8}} = 2^1 \cdot x^{\frac{1}{8}} = 2 \cdot x^{\frac{1}{8}}
\][/tex]
2. Combine the simplified term components:
- Now, combining these simplified terms with the constant 6 from the original expression, we get:
[tex]\[
6 \cdot (3 \cdot x^9) \cdot (2 \cdot x^{\frac{1}{8}})
\][/tex]
- First, handle the coefficients (numerical values):
[tex]\[
6 \cdot 3 \cdot 2 = 36
\][/tex]
- Next, handle the [tex]\(x\)[/tex] terms by combining the exponents:
[tex]\[
x^9 \cdot x^{\frac{1}{8}} = x^{9 + \frac{1}{8}}
\][/tex]
- Adding the exponents together:
[tex]\[
9 + \frac{1}{8} = 9.125
\][/tex]
3. Forming the final expression:
- The combined expression is:
[tex]\[
36 \cdot x^{9.125}
\][/tex]
Which means in the form [tex]\(a x^b\)[/tex], [tex]\(a = 36\)[/tex] and [tex]\(b = 9.125\)[/tex].
4. Calculating [tex]\(a + b\)[/tex]:
[tex]\[
a + b = 36 + 9.125 = 45.125
\][/tex]
Therefore, the value of [tex]\(a + b\)[/tex] is [tex]\(45.125\)[/tex].
