Multiplying Monomials | PPT - Free Printable

The task involves multiplying monomials, which are algebraic expressions consisting of a coefficient and one or more variables raised to powers. The key rules for multiplying monomials are:

1. Multiply the coefficients: Combine the numerical parts of the monomials.
2. Add the exponents of like bases: When multiplying variables with the same base, add their exponents.

Let's solve each problem step by step:

---

Problem 1:

$$
(4a^7)(2a^6)
$$

#### Step 1: Multiply the coefficients
The coefficients are 4 and 2. Their product is:
$$
4 \times 2 = 8
$$

#### Step 2: Add the exponents of the variable \(a\)
The exponents of \(a\) are 7 and 6. Adding them gives:
$$
7 + 6 = 13
$$

#### Final Answer:
Combine the results:
$$
(4a^7)(2a^6) = 8a^{13}
$$

---

Problem 2:

$$
(7w^2)(10w^{10})
$$

#### Step 1: Multiply the coefficients
The coefficients are 7 and 10. Their product is:
$$
7 \times 10 = 70
$$

#### Step 2: Add the exponents of the variable \(w\)
The exponents of \(w\) are 2 and 10. Adding them gives:
$$
2 + 10 = 12
$$

#### Final Answer:
Combine the results:
$$
(7w^2)(10w^{10}) = 70w^{12}
$$

---

Problem 3:

$$
(2r)(-3r^5)
$$

#### Step 1: Multiply the coefficients
The coefficients are 2 and -3. Their product is:
$$
2 \times (-3) = -6
$$

#### Step 2: Add the exponents of the variable \(r\)
The exponents of \(r\) are 1 (since \(r = r^1\)) and 5. Adding them gives:
$$
1 + 5 = 6
$$

#### Final Answer:
Combine the results:
$$
(2r)(-3r^5) = -6r^6
$$

---

Problem 4:

$$
(3k^4)(7k^4)
$$

#### Step 1: Multiply the coefficients
The coefficients are 3 and 7. Their product is:
$$
3 \times 7 = 21
$$

#### Step 2: Add the exponents of the variable \(k\)
The exponents of \(k\) are 4 and 4. Adding them gives:
$$
4 + 4 = 8
$$

#### Final Answer:
Combine the results:
$$
(3k^4)(7k^4) = 21k^8
$$

---

Problem 5:

$$
(12x^2)(2y^2)
$$

#### Step 1: Multiply the coefficients
The coefficients are 12 and 2. Their product is:
$$
12 \times 2 = 24
$$

#### Step 2: Add the exponents of the variables \(x\) and \(y\)
- For \(x\), the exponent is 2, and there is no other \(x\) term, so it remains \(x^2\).
- For \(y\), the exponent is 2, and there is no other \(y\) term, so it remains \(y^2\).

#### Final Answer:
Combine the results:
$$
(12x^2)(2y^2) = 24x^2y^2
$$

---

Final Answers:

1. $$
\boxed{8a^{13}}
$$
2. $$
\boxed{70w^{12}}
$$
3. $$
\boxed{-6r^6}
$$
4. $$
\boxed{21k^8}
$$
5. $$
\boxed{24x^2y^2}
$$

These solutions follow the rules of multiplying monomials: multiply the coefficients and add the exponents of like bases.