Exponent Multiplication Worksheets - 15 Worksheets Library - Free Printable

Problem: Multiplying Exponents

The task is to solve the given expressions by multiplying exponents and writing the answers in only positive exponents. We will use the following rules of exponents:
1. Product Rule: When multiplying powers with the same base, add the exponents: \( a^m \cdot a^n = a^{m+n} \).
2. Negative Exponent Rule: Convert negative exponents to positive by moving the term to the denominator (or vice versa): \( a^{-n} = \frac{1}{a^n} \).

Let's solve each problem step by step.

---

Problem 1: \( 3a^2b^3 \cdot 2a^3b^3 \)

1. Multiply the coefficients: \( 3 \cdot 2 = 6 \).
2. Add the exponents of \( a \): \( a^2 \cdot a^3 = a^{2+3} = a^5 \).
3. Add the exponents of \( b \): \( b^3 \cdot b^3 = b^{3+3} = b^6 \).

Final answer: \( 6a^5b^6 \).

---

Problem 2: \( 3c^2k^{-3} \cdot 5c^{-7}k^4 \)

1. Multiply the coefficients: \( 3 \cdot 5 = 15 \).
2. Add the exponents of \( c \): \( c^2 \cdot c^{-7} = c^{2 + (-7)} = c^{-5} \).
3. Add the exponents of \( k \): \( k^{-3} \cdot k^4 = k^{-3 + 4} = k^1 = k \).

Since we need positive exponents, rewrite \( c^{-5} \) as \( \frac{1}{c^5} \):

Final answer: \( \frac{15k}{c^5} \).

---

Problem 3: \( -8z^5 \cdot 4z^2y^3 \)

1. Multiply the coefficients: \( -8 \cdot 4 = -32 \).
2. Add the exponents of \( z \): \( z^5 \cdot z^2 = z^{5+2} = z^7 \).
3. The exponent of \( y \) remains \( y^3 \) since there is no other \( y \)-term to multiply.

Final answer: \( -32z^7y^3 \).

---

Problem 4: \( a^2 \cdot a^{-4} \)

1. Add the exponents of \( a \): \( a^2 \cdot a^{-4} = a^{2 + (-4)} = a^{-2} \).

Since we need positive exponents, rewrite \( a^{-2} \) as \( \frac{1}{a^2} \):

Final answer: \( \frac{1}{a^2} \).

---

Problem 5: \( b^{-4}c^5 \cdot 4b^2c^{-8} \cdot 3b^3c^{-4} \)

1. Multiply the coefficients: \( 1 \cdot 4 \cdot 3 = 12 \).
2. Add the exponents of \( b \): \( b^{-4} \cdot b^2 \cdot b^3 = b^{-4 + 2 + 3} = b^1 = b \).
3. Add the exponents of \( c \): \( c^5 \cdot c^{-8} \cdot c^{-4} = c^{5 + (-8) + (-4)} = c^{-7} \).

Since we need positive exponents, rewrite \( c^{-7} \) as \( \frac{1}{c^7} \):

Final answer: \( \frac{12b}{c^7} \).

---

Problem 6: \( 5a^{-4} \cdot 2a^{-2} \cdot 3a \)

1. Multiply the coefficients: \( 5 \cdot 2 \cdot 3 = 30 \).
2. Add the exponents of \( a \): \( a^{-4} \cdot a^{-2} \cdot a^1 = a^{-4 + (-2) + 1} = a^{-5} \).

Since we need positive exponents, rewrite \( a^{-5} \) as \( \frac{1}{a^5} \):

Final answer: \( \frac{30}{a^5} \).

---

Problem 7: \( 4x^4y^3 \cdot 2x^{-5}y^{-7} \cdot 5x^1y^7 \)

1. Multiply the coefficients: \( 4 \cdot 2 \cdot 5 = 40 \).
2. Add the exponents of \( x \): \( x^4 \cdot x^{-5} \cdot x^1 = x^{4 + (-5) + 1} = x^0 = 1 \).
3. Add the exponents of \( y \): \( y^3 \cdot y^{-7} \cdot y^7 = y^{3 + (-7) + 7} = y^3 \).

Final answer: \( 40y^3 \).

---

Problem 8: \( 4z^5w^{-6} \cdot 4z^{-2}w^2 \)

1. Multiply the coefficients: \( 4 \cdot 4 = 16 \).
2. Add the exponents of \( z \): \( z^5 \cdot z^{-2} = z^{5 + (-2)} = z^3 \).
3. Add the exponents of \( w \): \( w^{-6} \cdot w^2 = w^{-6 + 2} = w^{-4} \).

Since we need positive exponents, rewrite \( w^{-4} \) as \( \frac{1}{w^4} \):

Final answer: \( \frac{16z^3}{w^4} \).

---

Problem 9: \( 3a^2 \cdot 2a^3b^4 \cdot 4a^7b^2 \)

1. Multiply the coefficients: \( 3 \cdot 2 \cdot 4 = 24 \).
2. Add the exponents of \( a \): \( a^2 \cdot a^3 \cdot a^7 = a^{2 + 3 + 7} = a^{12} \).
3. Add the exponents of \( b \): \( b^4 \cdot b^2 = b^{4 + 2} = b^6 \).

Final answer: \( 24a^{12}b^6 \).

---

Problem 10: \( 3x^5y^{-3} \cdot 2x^2y^4 \cdot 4x^7 \)

1. Multiply the coefficients: \( 3 \cdot 2 \cdot 4 = 24 \).
2. Add the exponents of \( x \): \( x^5 \cdot x^2 \cdot x^7 = x^{5 + 2 + 7} = x^{14} \).
3. Add the exponents of \( y \): \( y^{-3} \cdot y^4 = y^{-3 + 4} = y^1 = y \).

Final answer: \( 24x^{14}y \).

---

Final Answers

1. \( 6a^5b^6 \)
2. \( \frac{15k}{c^5} \)
3. \( -32z^7y^3 \)
4. \( \frac{1}{a^2} \)
5. \( \frac{12b}{c^7} \)
6. \( \frac{30}{a^5} \)
7. \( 40y^3 \)
8. \( \frac{16z^3}{w^4} \)
9. \( 24a^{12}b^6 \)
10. \( 24x^{14}y \)

\[
\boxed{
\begin{array}{ll}
1. & 6a^5b^6 \\
2. & \frac{15k}{c^5} \\
3. & -32z^7y^3 \\
4. & \frac{1}{a^2} \\
5. & \frac{12b}{c^7} \\
6. & \frac{30}{a^5} \\
7. & 40y^3 \\
8. & \frac{16z^3}{w^4} \\
9. & 24a^{12}b^6 \\
10. & 24x^{14}y \\
\end{array}
}
\]