Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

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.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.