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Factor:
[tex]\[ 7a(5a-1)^{-\frac{5}{4}} + 2(5a-1)^{-\frac{1}{4}} \][/tex]

Select the correct answer below:

A. [tex]\((5a-1)^{-\frac{1}{4}}(14a-2)\)[/tex]

B. [tex]\((5a-1)^{-\frac{5}{4}}(17a-2)\)[/tex]

C. [tex]\((5a+1)^{-\frac{1}{4}}(17a-2)\)[/tex]

D. [tex]\((5a-1)^{-\frac{5}{4}}(17a+2)\)[/tex]


Sagot :

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.
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