Answer :
To simplify the expression [tex]\(14 x^5 (13 x^2 + 13 x^5)\)[/tex], we will follow these steps carefully:
1. Distribute the [tex]\(14 x^5\)[/tex] through the parentheses:
[tex]\[
14 x^5 \times (13 x^2 + 13 x^5)
\][/tex]
This means you will multiply [tex]\(14 x^5\)[/tex] by each term inside the parentheses.
2. Multiply [tex]\(14 x^5\)[/tex] by [tex]\(13 x^2\)[/tex]:
[tex]\[
14 x^5 \times 13 x^2 = 182 x^{5+2} = 182 x^7
\][/tex]
3. Multiply [tex]\(14 x^5\)[/tex] by [tex]\(13 x^5\)[/tex]:
[tex]\[
14 x^5 \times 13 x^5 = 182 x^{5+5} = 182 x^{10}
\][/tex]
4. Combine the results from steps 2 and 3:
[tex]\[
182 x^7 + 182 x^{10}
\][/tex]
Now, let's match this simplified expression to the given options:
- Option a: [tex]\(27 x^{10} + 27 x^{25}\)[/tex]
- Option b: [tex]\(182 x^{10} + 13 x^5\)[/tex]
- Option c: [tex]\(182 x^7 + 182 x^{10}\)[/tex]
- Option d: [tex]\(27 x^7 + 27 x^{10}\)[/tex]
The correct answer is [tex]\(182 x^7 + 182 x^{10}\)[/tex], which matches Option c.
1. Distribute the [tex]\(14 x^5\)[/tex] through the parentheses:
[tex]\[
14 x^5 \times (13 x^2 + 13 x^5)
\][/tex]
This means you will multiply [tex]\(14 x^5\)[/tex] by each term inside the parentheses.
2. Multiply [tex]\(14 x^5\)[/tex] by [tex]\(13 x^2\)[/tex]:
[tex]\[
14 x^5 \times 13 x^2 = 182 x^{5+2} = 182 x^7
\][/tex]
3. Multiply [tex]\(14 x^5\)[/tex] by [tex]\(13 x^5\)[/tex]:
[tex]\[
14 x^5 \times 13 x^5 = 182 x^{5+5} = 182 x^{10}
\][/tex]
4. Combine the results from steps 2 and 3:
[tex]\[
182 x^7 + 182 x^{10}
\][/tex]
Now, let's match this simplified expression to the given options:
- Option a: [tex]\(27 x^{10} + 27 x^{25}\)[/tex]
- Option b: [tex]\(182 x^{10} + 13 x^5\)[/tex]
- Option c: [tex]\(182 x^7 + 182 x^{10}\)[/tex]
- Option d: [tex]\(27 x^7 + 27 x^{10}\)[/tex]
The correct answer is [tex]\(182 x^7 + 182 x^{10}\)[/tex], which matches Option c.
