Exponent Multiplication Worksheets - 15 Worksheets Library - Free Printable

Explanation:
We are multiplying expressions with exponents. The rule we use is:

- When multiplying like bases, add the exponents:
$ a^m \cdot a^n = a^{m+n} $

- Also, multiply the coefficients (the numbers in front) separately.

And the problem says: write the answer using only positive exponents — so if any exponent ends up negative, we need to rewrite it (but in this worksheet, all final answers will end up with positive exponents after simplifying).

Let’s go one by one:

---

1) $ 3a^2b^3 \cdot 2a^5b^3 $
Coefficients: $ 3 \cdot 2 = 6 $
$ a $-terms: $ a^2 \cdot a^5 = a^{2+5} = a^7 $
$ b $-terms: $ b^3 \cdot b^3 = b^{3+3} = b^6 $
→ Answer: $ 6a^7b^6 $

---

2) $ 3c^5k^3 \cdot 5c^{-7}k^4 $
Coefficients: $ 3 \cdot 5 = 15 $
$ c $-terms: $ c^5 \cdot c^{-7} = c^{5 + (-7)} = c^{-2} $
But we need only positive exponents, so $ c^{-2} = \frac{1}{c^2} $. However, the instruction says “write the answer in only positive exponents” — but since this is multiplication of monomials, and no division is shown, maybe they expect us to simplify fully *as a monomial*, meaning if exponent is negative, we should avoid it — but wait! Let's double-check: maybe we made a mistake? No — the problem includes negative exponents in the input (like $ c^{-7} $), so we must handle them.

But the instruction says: *“write the answer in only positive exponents.”* So we cannot leave $ c^{-2} $. That means we must express the result as a fraction? But all other problems seem to result in positive exponents. Let’s keep going — maybe later terms cancel negatives.

Wait — actually, re-read: It says “Solve the following exponents and write the answer in only positive exponents.” So if the final exponent is negative, we need to rewrite it as a positive exponent in the denominator — but since all answers are expected to be monomials (no fractions shown in the layout), perhaps the problems are designed so that all final exponents are positive. Let’s verify each carefully.

In #2:
$ c^5 \cdot c^{-7} = c^{-2} $ → negative. Hmm. But maybe the problem expects us to write it as $ \frac{15k^7}{c^2} $? However, the format of the worksheet shows blank boxes for answers — likely expecting monomial form. Let’s hold and check others; maybe we misread.

Wait — look at problem 10: $ 3x^5y^{-3} \cdot 2x^3y^{-4} \cdot 4x^{-7} $
Exponents:
x: $ 5 + 3 + (-7) = 1 $
y: $ -3 + (-4) = -7 $ → still negative.

So clearly, some answers will have negative exponents unless we convert. But instruction says: *write the answer in only positive exponents*. So we must eliminate negative exponents — meaning express as fraction if needed. But typical middle-school worksheets like this often expect you to combine and if exponent is negative, write as denominator.

However, looking at the source (15Worksheets.com), and standard practice for such worksheets: they usually design problems so that after combining, all exponents are non-negative. Let me double-check #2:

$ 3c^5k^3 \cdot 5c^{-7}k^4 $
= $ (3\cdot5)(c^{5-7})(k^{3+4}) = 15 c^{-2} k^7 $

To write with only positive exponents: $ \frac{15k^7}{c^2} $

But the answer box is a single line — maybe they accept that form. Let’s assume we are allowed to write as fraction when needed.

However, let’s check if any problem has variables with negative total exponent — yes, #10 does. So we’ll need to write fractions for those.

But wait — maybe the instruction means “simplify, and if any exponent is negative, rewrite the term so the exponent becomes positive” — i.e., move to denominator.

Let’s proceed and do all, converting negatives to positive via denominator.

---

3) $ -8z^5 \cdot 4z^2y^3 $
Coefficients: $ -8 \cdot 4 = -32 $
$ z $: $ z^5 \cdot z^2 = z^7 $
$ y $: only $ y^3 $
→ $ -32z^7y^3 $ ✔ all positive exponents.

---

4) $ a^2 \cdot a^4 = a^{6} $
→ $ a^6 $

---

5) $ b^4c^5 \cdot 4b^2c^{-8} \cdot 3b^3c^4 $
Coefficients: $ 1 \cdot 4 \cdot 3 = 12 $
$ b $: $ b^4 \cdot b^2 \cdot b^3 = b^{4+2+3} = b^9 $
$ c $: $ c^5 \cdot c^{-8} \cdot c^4 = c^{5 - 8 + 4} = c^{1} $
→ $ 12b^9c $

---

6) $ 5a^4 \cdot 2a^2 \cdot 3a $
Coefficients: $ 5 \cdot 2 \cdot 3 = 30 $
$ a $: $ a^4 \cdot a^2 \cdot a^1 = a^{7} $
→ $ 30a^7 $

---

7) $ 4x^4y^3 \cdot 2x^5y^{-7} \cdot 5x^3y^2 $
Coefficients: $ 4 \cdot 2 \cdot 5 = 40 $
$ x $: $ x^{4+5+3} = x^{12} $
$ y $: $ y^{3 + (-7) + 2} = y^{-2} $
→ $ 40x^{12}y^{-2} = \frac{40x^{12}}{y^2} $

---

8) $ 4z^5w^6 \cdot 4z^3w^2 $
Coefficients: $ 4 \cdot 4 = 16 $
$ z $: $ z^{5+3} = z^8 $
$ w $: $ w^{6+2} = w^8 $
→ $ 16z^8w^8 $

---

9) $ 3a^2 \cdot 2a^3b^4 \cdot 4a^7b^2 $
Coefficients: $ 3 \cdot 2 \cdot 4 = 24 $
$ a $: $ a^{2+3+7} = a^{12} $
$ b $: $ b^{4+2} = b^6 $
→ $ 24a^{12}b^6 $

---

10) $ 3x^5y^{-3} \cdot 2x^3y^{-4} \cdot 4x^{-7} $
Coefficients: $ 3 \cdot 2 \cdot 4 = 24 $
$ x $: $ x^{5+3+(-7)} = x^{1} $
$ y $: $ y^{-3 + (-4)} = y^{-7} $
→ $ 24x y^{-7} = \frac{24x}{y^7} $

Now, confirm all:

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

These are simplified with only positive exponents.

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