Answer :
Sure! Let's go through each expression step-by-step and apply the product rule for exponents, which states that when you multiply two powers with the same base, you add the exponents.
31. [tex]\(x^2 \cdot x^5\)[/tex]
- Add the exponents: [tex]\(2 + 5 = 7\)[/tex]
- Result: [tex]\(x^7\)[/tex]
32. [tex]\(y^2 \cdot y\)[/tex]
- Remember [tex]\(y\)[/tex] is the same as [tex]\(y^1\)[/tex].
- Add the exponents: [tex]\(2 + 1 = 3\)[/tex]
- Result: [tex]\(y^3\)[/tex]
33. [tex]\((-3)^3 \cdot (-3)^9\)[/tex]
- The bases are the same, so add the exponents: [tex]\(3 + 9 = 12\)[/tex]
- Result: [tex]\((-3)^{12}\)[/tex]
34. [tex]\((-5)^7 \cdot (-5)^6\)[/tex]
- Add the exponents: [tex]\(7 + 6 = 13\)[/tex]
- Result: [tex]\((-5)^{13}\)[/tex]
35. [tex]\((5y^4)(3y)\)[/tex]
- Multiply the coefficients: [tex]\(5 \times 3 = 15\)[/tex]
- For the [tex]\(y\)[/tex] term: add the exponents: [tex]\(4 + 1 = 5\)[/tex]
- Result: [tex]\(15y^5\)[/tex]
36. [tex]\((-2z^3)(-2z^2)\)[/tex]
- Multiply the coefficients: [tex]\((-2) \times (-2) = 4\)[/tex]
- For the [tex]\(z\)[/tex] term: add the exponents: [tex]\(3 + 2 = 5\)[/tex]
- Result: [tex]\(4z^5\)[/tex]
37. [tex]\((x^9 y)(x^{10} y^5)\)[/tex]
- For the [tex]\(x\)[/tex] term: add the exponents: [tex]\(9 + 10 = 19\)[/tex]
- For the [tex]\(y\)[/tex] term: add the exponents: [tex]\(1 + 5 = 6\)[/tex]
- Result: [tex]\(x^{19} y^6\)[/tex]
38. [tex]\((a^2 b)(a^{13} b^{17})\)[/tex]
- For the [tex]\(a\)[/tex] term: add the exponents: [tex]\(2 + 13 = 15\)[/tex]
- For the [tex]\(b\)[/tex] term: add the exponents: [tex]\(1 + 17 = 18\)[/tex]
- Result: [tex]\(a^{15} b^{18}\)[/tex]
39. [tex]\((-8mn^6)(9m^2n^2)\)[/tex]
- Multiply the coefficients: [tex]\((-8) \times 9 = -72\)[/tex]
- For the [tex]\(m\)[/tex] term: add the exponents: [tex]\(1 + 2 = 3\)[/tex]
- For the [tex]\(n\)[/tex] term: add the exponents: [tex]\(6 + 2 = 8\)[/tex]
- Result: [tex]\(-72m^3n^8\)[/tex]
40. [tex]\((-7a^3b^3)(7a^{19}b)\)[/tex]
- Multiply the coefficients: [tex]\((-7) \times 7 = -49\)[/tex]
- For the [tex]\(a\)[/tex] term: add the exponents: [tex]\(3 + 19 = 22\)[/tex]
- For the [tex]\(b\)[/tex] term: add the exponents: [tex]\(3 + 1 = 4\)[/tex]
- Result: [tex]\(-49a^{22}b^4\)[/tex]
41. [tex]\((4z^{10})(-6z^7)(z^3)\)[/tex]
- Multiply the coefficients: [tex]\(4 \times (-6) \times 1 = -24\)[/tex]
- For the [tex]\(z\)[/tex] term: add the exponents: [tex]\(10 + 7 + 3 = 20\)[/tex]
- Result: [tex]\(-24z^{20}\)[/tex]
42. [tex]\((12x^5)(-x^6)(x^4)\)[/tex]
- Multiply the coefficients: [tex]\(12 \times (-1) \times 1 = -12\)[/tex]
- For the [tex]\(x\)[/tex] term: add the exponents: [tex]\(5 + 6 + 4 = 15\)[/tex]
- Result: [tex]\(-12x^{15}\)[/tex]
31. [tex]\(x^2 \cdot x^5\)[/tex]
- Add the exponents: [tex]\(2 + 5 = 7\)[/tex]
- Result: [tex]\(x^7\)[/tex]
32. [tex]\(y^2 \cdot y\)[/tex]
- Remember [tex]\(y\)[/tex] is the same as [tex]\(y^1\)[/tex].
- Add the exponents: [tex]\(2 + 1 = 3\)[/tex]
- Result: [tex]\(y^3\)[/tex]
33. [tex]\((-3)^3 \cdot (-3)^9\)[/tex]
- The bases are the same, so add the exponents: [tex]\(3 + 9 = 12\)[/tex]
- Result: [tex]\((-3)^{12}\)[/tex]
34. [tex]\((-5)^7 \cdot (-5)^6\)[/tex]
- Add the exponents: [tex]\(7 + 6 = 13\)[/tex]
- Result: [tex]\((-5)^{13}\)[/tex]
35. [tex]\((5y^4)(3y)\)[/tex]
- Multiply the coefficients: [tex]\(5 \times 3 = 15\)[/tex]
- For the [tex]\(y\)[/tex] term: add the exponents: [tex]\(4 + 1 = 5\)[/tex]
- Result: [tex]\(15y^5\)[/tex]
36. [tex]\((-2z^3)(-2z^2)\)[/tex]
- Multiply the coefficients: [tex]\((-2) \times (-2) = 4\)[/tex]
- For the [tex]\(z\)[/tex] term: add the exponents: [tex]\(3 + 2 = 5\)[/tex]
- Result: [tex]\(4z^5\)[/tex]
37. [tex]\((x^9 y)(x^{10} y^5)\)[/tex]
- For the [tex]\(x\)[/tex] term: add the exponents: [tex]\(9 + 10 = 19\)[/tex]
- For the [tex]\(y\)[/tex] term: add the exponents: [tex]\(1 + 5 = 6\)[/tex]
- Result: [tex]\(x^{19} y^6\)[/tex]
38. [tex]\((a^2 b)(a^{13} b^{17})\)[/tex]
- For the [tex]\(a\)[/tex] term: add the exponents: [tex]\(2 + 13 = 15\)[/tex]
- For the [tex]\(b\)[/tex] term: add the exponents: [tex]\(1 + 17 = 18\)[/tex]
- Result: [tex]\(a^{15} b^{18}\)[/tex]
39. [tex]\((-8mn^6)(9m^2n^2)\)[/tex]
- Multiply the coefficients: [tex]\((-8) \times 9 = -72\)[/tex]
- For the [tex]\(m\)[/tex] term: add the exponents: [tex]\(1 + 2 = 3\)[/tex]
- For the [tex]\(n\)[/tex] term: add the exponents: [tex]\(6 + 2 = 8\)[/tex]
- Result: [tex]\(-72m^3n^8\)[/tex]
40. [tex]\((-7a^3b^3)(7a^{19}b)\)[/tex]
- Multiply the coefficients: [tex]\((-7) \times 7 = -49\)[/tex]
- For the [tex]\(a\)[/tex] term: add the exponents: [tex]\(3 + 19 = 22\)[/tex]
- For the [tex]\(b\)[/tex] term: add the exponents: [tex]\(3 + 1 = 4\)[/tex]
- Result: [tex]\(-49a^{22}b^4\)[/tex]
41. [tex]\((4z^{10})(-6z^7)(z^3)\)[/tex]
- Multiply the coefficients: [tex]\(4 \times (-6) \times 1 = -24\)[/tex]
- For the [tex]\(z\)[/tex] term: add the exponents: [tex]\(10 + 7 + 3 = 20\)[/tex]
- Result: [tex]\(-24z^{20}\)[/tex]
42. [tex]\((12x^5)(-x^6)(x^4)\)[/tex]
- Multiply the coefficients: [tex]\(12 \times (-1) \times 1 = -12\)[/tex]
- For the [tex]\(x\)[/tex] term: add the exponents: [tex]\(5 + 6 + 4 = 15\)[/tex]
- Result: [tex]\(-12x^{15}\)[/tex]
