Multiply Polynomials Worksheet featuring 14 algebraic expressions to multiply and simplify.

To solve the given problems, we need to multiply the polynomials and simplify the results. The general steps are:

1. Multiply the coefficients (the numerical parts).
2. Add the exponents of like bases (use the property \(a^m \cdot a^n = a^{m+n}\)).

Let's solve each problem step by step.

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Problem 1: \( (3x^5)(7x^6) \)

- Multiply the coefficients: \(3 \cdot 7 = 21\).
- Add the exponents of \(x\): \(x^5 \cdot x^6 = x^{5+6} = x^{11}\).

So, the result is:
\[
(3x^5)(7x^6) = 21x^{11}
\]

---

Problem 2: \( (-6a^4)(4a^4) \)

- Multiply the coefficients: \(-6 \cdot 4 = -24\).
- Add the exponents of \(a\): \(a^4 \cdot a^4 = a^{4+4} = a^8\).

So, the result is:
\[
(-6a^4)(4a^4) = -24a^8
\]

---

Problem 3: \( (-8t^4)(-7t^2) \)

- Multiply the coefficients: \(-8 \cdot -7 = 56\).
- Add the exponents of \(t\): \(t^4 \cdot t^2 = t^{4+2} = t^6\).

So, the result is:
\[
(-8t^4)(-7t^2) = 56t^6
\]

---

Problem 4: \( (-9x^3)(7x^5) \)

- Multiply the coefficients: \(-9 \cdot 7 = -63\).
- Add the exponents of \(x\): \(x^3 \cdot x^5 = x^{3+5} = x^8\).

So, the result is:
\[
(-9x^3)(7x^5) = -63x^8
\]

---

Problem 5: \( (-6x^4)(-3x^2) \)

- Multiply the coefficients: \(-6 \cdot -3 = 18\).
- Add the exponents of \(x\): \(x^4 \cdot x^2 = x^{4+2} = x^6\).

So, the result is:
\[
(-6x^4)(-3x^2) = 18x^6
\]

---

Problem 6: \( (-7n^3)(-3n^6) \)

- Multiply the coefficients: \(-7 \cdot -3 = 21\).
- Add the exponents of \(n\): \(n^3 \cdot n^6 = n^{3+6} = n^9\).

So, the result is:
\[
(-7n^3)(-3n^6) = 21n^9
\]

---

Problem 7: \( (6d^4)(-10d^5) \)

- Multiply the coefficients: \(6 \cdot -10 = -60\).
- Add the exponents of \(d\): \(d^4 \cdot d^5 = d^{4+5} = d^9\).

So, the result is:
\[
(6d^4)(-10d^5) = -60d^9
\]

---

Problem 8: \( (c^5)(8c^2) \)

- Multiply the coefficients: \(1 \cdot 8 = 8\) (since \(c^5\) has an implied coefficient of 1).
- Add the exponents of \(c\): \(c^5 \cdot c^2 = c^{5+2} = c^7\).

So, the result is:
\[
(c^5)(8c^2) = 8c^7
\]

---

Problem 9: \( (7n^5)(2n) \)

- Multiply the coefficients: \(7 \cdot 2 = 14\).
- Add the exponents of \(n\): \(n^5 \cdot n = n^{5+1} = n^6\).

So, the result is:
\[
(7n^5)(2n) = 14n^6
\]

---

Problem 10: \( (-2y^4)(8y^2) \)

- Multiply the coefficients: \(-2 \cdot 8 = -16\).
- Add the exponents of \(y\): \(y^4 \cdot y^2 = y^{4+2} = y^6\).

So, the result is:
\[
(-2y^4)(8y^2) = -16y^6
\]

---

Problem 11: \( (0t^5)(-b^3t^2) \)

- Multiply the coefficients: \(0 \cdot -b^3 = 0\) (any number multiplied by 0 is 0).
- The exponents do not matter because the result is 0.

So, the result is:
\[
(0t^5)(-b^3t^2) = 0
\]

---

Problem 12: \( (-6k^5n^2)(2k^2n^4) \)

- Multiply the coefficients: \(-6 \cdot 2 = -12\).
- Add the exponents of \(k\): \(k^5 \cdot k^2 = k^{5+2} = k^7\).
- Add the exponents of \(n\): \(n^2 \cdot n^4 = n^{2+4} = n^6\).

So, the result is:
\[
(-6k^5n^2)(2k^2n^4) = -12k^7n^6
\]

---

Problem 13: \( (-3p^2c)(-p^3c^2) \)

- Multiply the coefficients: \(-3 \cdot -1 = 3\).
- Add the exponents of \(p\): \(p^2 \cdot p^3 = p^{2+3} = p^5\).
- Add the exponents of \(c\): \(c \cdot c^2 = c^{1+2} = c^3\).

So, the result is:
\[
(-3p^2c)(-p^3c^2) = 3p^5c^3
\]

---

Problem 14: \( (-7m^3k^2)(10m^4k) \)

- Multiply the coefficients: \(-7 \cdot 10 = -70\).
- Add the exponents of \(m\): \(m^3 \cdot m^4 = m^{3+4} = m^7\).
- Add the exponents of \(k\): \(k^2 \cdot k = k^{2+1} = k^3\).

So, the result is:
\[
(-7m^3k^2)(10m^4k) = -70m^7k^3
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 21x^{11} \\
2. & \ -24a^8 \\
3. & \ 56t^6 \\
4. & \ -63x^8 \\
5. & \ 18x^6 \\
6. & \ 21n^9 \\
7. & \ -60d^9 \\
8. & \ 8c^7 \\
9. & \ 14n^6 \\
10. & \ -16y^6 \\
11. & \ 0 \\
12. & \ -12k^7n^6 \\
13. & \ 3p^5c^3 \\
14. & \ -70m^7k^3
\end{aligned}
}
\]