Answer :
Sure! Let's match each multiplication with its respective product by going through each expression step-by-step.
1. Expression 1:
Multiply: [tex]\(4x^5 \left(8x^4 + 3x - 9\right)\)[/tex].
Distributing [tex]\(4x^5\)[/tex] across each term inside the parentheses, we get:
- [tex]\(4x^5 \cdot 8x^4 = 32x^9\)[/tex]
- [tex]\(4x^5 \cdot 3x = 12x^6\)[/tex]
- [tex]\(4x^5 \cdot (-9) = -36x^5\)[/tex]
The product is:
[tex]\(32x^9 + 12x^6 - 36x^5\)[/tex].
2. Expression 2:
Multiply: [tex]\(2x^2 \left(10x^5 - 2x^3 - x\right)\)[/tex].
Distributing [tex]\(2x^2\)[/tex] across each term inside the parentheses, we get:
- [tex]\(2x^2 \cdot 10x^5 = 20x^7\)[/tex]
- [tex]\(2x^2 \cdot (-2x^3) = -4x^5\)[/tex]
- [tex]\(2x^2 \cdot (-x) = -2x^3\)[/tex]
The product is:
[tex]\(20x^7 - 4x^5 - 2x^3\)[/tex].
3. Expression 3:
Fraction: [tex]\(\frac{20x^7 - 4x^5 - 2x^3}{32x^9 + 12x^6 - 36x^5}\)[/tex].
This is already simplified in the form of a fraction, so no further simplification is needed.
The product is:
[tex]\(\frac{20x^7 - 4x^5 - 2x^3}{32x^9 + 12x^6 - 36x^5}\)[/tex].
4. Expression 4:
Multiply: [tex]\(-5x^2 \left(7x^5 - 4x^4 + 2x^3\right)\)[/tex].
Distributing [tex]\(-5x^2\)[/tex] across each term inside the parentheses, we get:
- [tex]\(-5x^2 \cdot 7x^5 = -35x^7\)[/tex]
- [tex]\(-5x^2 \cdot (-4x^4) = 20x^6\)[/tex]
- [tex]\(-5x^2 \cdot 2x^3 = -10x^5\)[/tex]
The product is:
[tex]\(-35x^7 + 20x^6 - 10x^5\)[/tex].
5. Expression 5:
Multiply: [tex]\(-3x^3 \left(x^6 - 11x^2 - 5\right)\)[/tex].
Distributing [tex]\(-3x^3\)[/tex] across each term inside the parentheses, we get:
- [tex]\(-3x^3 \cdot x^6 = -3x^9\)[/tex]
- [tex]\(-3x^3 \cdot (-11x^2) = 33x^5\)[/tex]
- [tex]\(-3x^3 \cdot (-5) = 15x^3\)[/tex]
The product is:
[tex]\(-3x^9 + 33x^5 + 15x^3\)[/tex].
So, each expression is matched with its product as calculated above.
1. Expression 1:
Multiply: [tex]\(4x^5 \left(8x^4 + 3x - 9\right)\)[/tex].
Distributing [tex]\(4x^5\)[/tex] across each term inside the parentheses, we get:
- [tex]\(4x^5 \cdot 8x^4 = 32x^9\)[/tex]
- [tex]\(4x^5 \cdot 3x = 12x^6\)[/tex]
- [tex]\(4x^5 \cdot (-9) = -36x^5\)[/tex]
The product is:
[tex]\(32x^9 + 12x^6 - 36x^5\)[/tex].
2. Expression 2:
Multiply: [tex]\(2x^2 \left(10x^5 - 2x^3 - x\right)\)[/tex].
Distributing [tex]\(2x^2\)[/tex] across each term inside the parentheses, we get:
- [tex]\(2x^2 \cdot 10x^5 = 20x^7\)[/tex]
- [tex]\(2x^2 \cdot (-2x^3) = -4x^5\)[/tex]
- [tex]\(2x^2 \cdot (-x) = -2x^3\)[/tex]
The product is:
[tex]\(20x^7 - 4x^5 - 2x^3\)[/tex].
3. Expression 3:
Fraction: [tex]\(\frac{20x^7 - 4x^5 - 2x^3}{32x^9 + 12x^6 - 36x^5}\)[/tex].
This is already simplified in the form of a fraction, so no further simplification is needed.
The product is:
[tex]\(\frac{20x^7 - 4x^5 - 2x^3}{32x^9 + 12x^6 - 36x^5}\)[/tex].
4. Expression 4:
Multiply: [tex]\(-5x^2 \left(7x^5 - 4x^4 + 2x^3\right)\)[/tex].
Distributing [tex]\(-5x^2\)[/tex] across each term inside the parentheses, we get:
- [tex]\(-5x^2 \cdot 7x^5 = -35x^7\)[/tex]
- [tex]\(-5x^2 \cdot (-4x^4) = 20x^6\)[/tex]
- [tex]\(-5x^2 \cdot 2x^3 = -10x^5\)[/tex]
The product is:
[tex]\(-35x^7 + 20x^6 - 10x^5\)[/tex].
5. Expression 5:
Multiply: [tex]\(-3x^3 \left(x^6 - 11x^2 - 5\right)\)[/tex].
Distributing [tex]\(-3x^3\)[/tex] across each term inside the parentheses, we get:
- [tex]\(-3x^3 \cdot x^6 = -3x^9\)[/tex]
- [tex]\(-3x^3 \cdot (-11x^2) = 33x^5\)[/tex]
- [tex]\(-3x^3 \cdot (-5) = 15x^3\)[/tex]
The product is:
[tex]\(-3x^9 + 33x^5 + 15x^3\)[/tex].
So, each expression is matched with its product as calculated above.