Sure, let's walk through the steps to factor the expression [tex]\(7 a(5 a-1)^{-\frac{5}{4}}+2(5 a-1)^{-\frac{1}{4}}\)[/tex].
### Step-by-Step Solution:
1. Identify Common Factors:
The given expression is:
[tex]\[
7 a(5 a-1)^{-\frac{5}{4}}+2(5 a-1)^{-\frac{1}{4}}
\][/tex]
We notice that [tex]\( (5a - 1)^{-\frac{5}{4}} \)[/tex] and [tex]\( (5a - 1)^{-\frac{1}{4}} \)[/tex] have a common base of [tex]\( (5a - 1) \)[/tex].
2. Factor Out the Common Term:
The term with the highest negative exponent, [tex]\((5 a - 1)^{-\frac{5}{4}}\)[/tex], can be factored out:
[tex]\[
7 a (5 a-1)^{-\frac{5}{4}} + 2 (5 a-1)^{-\frac{1}{4}}
\][/tex]
Factoring out [tex]\(( (5 a-1)^{-\frac{5}{4}} ) we get:
\[
(5 a - 1)^{-\frac{5}{4}} \left[ 7a + 2 (5 a - 1)^{\frac{4}{4} - \frac{5}{4}} \right]
\]
3. Simplify the Expression Inside the Brackets:
Now, simplify the exponent calculation inside the brackets:
\[
(5 a - 1)^{-\frac{5}{4}} \left[ 7a + 2 (5 a - 1)^{-\frac{1}{4}} \right]
\]
Since \((5 a - 1)^{-\frac{1}{4}}\)[/tex] can be simplified:
[tex]\[
2(5 a - 1)^{-\frac{4}{4}+\frac{4}{4}}=2(5 a -1)^{\frac{0}{4}}=2(1)= 2
\][/tex]
Using the initial format, the complete expression under the brackets is:
[tex]\[
\left[ 7a + 2 \right]
\][/tex]
4. Final Expression:
Combining the simplified expression, we get:
[tex]\[
(5 a-1)^{-\frac{5}{4}} \cdot [7a + 2(5a-1)]
\][/tex]
Now multiply through:
[tex]\[
(5 a-1)^{-\frac{5}{4}} \cdot7a + 2
The above factorization is simplified to:
\left( \frac{17a - 2}{(5 a-1)^{\frac{5}{4}}}
As shown in simplified expression provided:
\frac{(17 a -2)}{(5 a-1)^{5/4}}
Hence, the correct answer:
$(5 a-1)^{-\frac{5}{4}}(17 a -2)$
### Answer:
\[
(5 a-1)^{-\frac{5}{4}}(17 a -2)
\][/tex]
This matches our solution steps to simplify and obtain a correct factorization of the initial expression.
