Laws of Exponents worksheet - Free Printable

Let’s solve each expression step by step, making sure all exponents are positive. We’ll use the laws of exponents:

- When multiplying powers with the same base: add exponents → $a^m \cdot a^n = a^{m+n}$
- When dividing powers with the same base: subtract exponents → $\frac{a^m}{a^n} = a^{m-n}$
- Negative exponent means flip to denominator: $a^{-n} = \frac{1}{a^n}$
- Power of a power: $(a^m)^n = a^{m \cdot n}$
- Coefficients (numbers) multiply or divide normally.

---

1. $9c^5 \cdot 5c^{-5}$

Multiply coefficients: $9 \cdot 5 = 45$

Add exponents for $c$: $5 + (-5) = 0$, so $c^0 = 1$

Solution: $45 \cdot 1 = 45$

→ Entire solution is purple

---

2. $2k^6g^2 \cdot 3kg^3$

Coefficients: $2 \cdot 3 = 6$

$k^6 \cdot k = k^{6+1} = k^7$

$g^2 \cdot g^3 = g^{2+3} = g^5$

Solution: $6k^7g^5$

→ Coefficient (6) = yellow; Exponent of $g$ (5) = dark blue

---

3. $7 \cdot 7^{-5}$

Same base: $7^1 \cdot 7^{-5} = 7^{1 + (-5)} = 7^{-4}$

Make exponent positive: $\frac{1}{7^4}$

Exponent in denominator is 4

→ Color red

---

4. $\frac{5h^2}{8h^4}$

Coefficients: $\frac{5}{8}$ — stays as fraction

$h^2 / h^4 = h^{2-4} = h^{-2} = \frac{1}{h^2}$

So overall: $\frac{5}{8h^2}$

Coefficient in denominator: 8 → pink

Exponent of $h$: 2 → light blue

---

5. $\frac{5s^{-2}n^{-5}}{7sn^{-6}}$

Break it down:

Coefficients: $\frac{5}{7}$

$s^{-2} / s^1 = s^{-2 - 1} = s^{-3} = \frac{1}{s^3}$

$n^{-5} / n^{-6} = n^{-5 - (-6)} = n^{1}$

So numerator has: $n^1$ → variable in numerator is n

→ Color purple

---

6. $\frac{5w^2}{9w^6y^4}$

Coefficients: $\frac{5}{9}$

$w^2 / w^6 = w^{2-6} = w^{-4} = \frac{1}{w^4}$

$y^4$ stays in denominator

So: $\frac{5}{9w^4y^4}$

Sum of coefficient in denominator (9) and exponent of $w$ (4): $9 + 4 = 13$

→ Color dark blue

---

7. $\frac{6^{-3}}{6}$

Write as: $6^{-3} \div 6^1 = 6^{-3 - 1} = 6^{-4} = \frac{1}{6^4}$

Denominator simplified: $6^4 = 1296$? Wait — but instruction says “denominator simplified” — probably just the exponent part? Let’s read again.

Actually, the expression simplifies to $\frac{1}{6^4}$, so denominator is $6^4$. But maybe they mean the exponent value? The instruction says “Denominator simplified (red)” — likely meaning the exponent in the denominator after simplifying.

Wait — let's recompute:

$\frac{6^{-3}}{6} = 6^{-3} \cdot 6^{-1} = 6^{-4} = \frac{1}{6^4}$

So denominator is $6^4$, which is 1296 — but that seems too big. Maybe they want the exponent? The coloring instruction says “Denominator simplified (red)” — perhaps they mean the exponent value in the denominator? Looking at other problems, sometimes they refer to the exponent number.

But in problem 3, they said “exponent in denominator (red)” and we used 4.

Here, denominator is $6^4$, so exponent is 4 → color red

Alternatively, if they mean the entire denominator value, it’s 1296 — but that doesn’t match any small numbers. Probably they mean the exponent.

Looking back at problem 3: $7 \cdot 7^{-5} = \frac{1}{7^4}$ → exponent in denominator is 4 → red.

Similarly here: $\frac{1}{6^4}$ → exponent in denominator is 4 → red.

Yes, consistent.

→ Color red

---

8. $\left(\frac{1}{n}\right)^6 \cdot \left(\frac{1}{n}\right)^3 \cdot \left(\frac{1}{n}\right)^5$

This is $\frac{1}{n^6} \cdot \frac{1}{n^3} \cdot \frac{1}{n^5} = \frac{1}{n^{6+3+5}} = \frac{1}{n^{14}}$

Power of the base: the base is $\frac{1}{n}$, and we’re raising it to total power 14? Or the exponent on n?

The expression is written as product of three terms, each being $(\frac{1}{n})^k$, so combined it’s $(\frac{1}{n})^{14}$

“Power of the base” — base is $\frac{1}{n}$, power is 14.

But coloring instruction says “Power of the base (light blue)” — so 14? That seems large. Maybe they mean the exponent on n in the final answer? Which is 14.

But let’s see: in the simplified form $\frac{1}{n^{14}}$, the exponent of n is 14.

Perhaps “power of the base” refers to the total exponent applied to the base $\frac{1}{n}$, which is 14.

I think it’s 14 → color light blue

---

9. $bs \cdot 8b^3s^5$

Coefficients: 1 * 8 = 8

$b^1 \cdot b^3 = b^4$

$s^1 \cdot s^5 = s^6$

Solution: $8b^4s^6$

Sum of exponents of b and s: 4 + 6 = 10

→ Color yellow

---

10. $\frac{6d^{-5}h^2}{5d^3h^{-3}}$

Coefficients: $\frac{6}{5}$

$d^{-5} / d^3 = d^{-5-3} = d^{-8} = \frac{1}{d^8}$

$h^2 / h^{-3} = h^{2 - (-3)} = h^5$

So: $\frac{6h^5}{5d^8}$

Variable in denominator: d

→ Color pink

---

11. $\frac{2^4}{2^6}$

$2^{4-6} = 2^{-2} = \frac{1}{2^2}$

Numerator is 1 (since it’s $\frac{1}{4}$, numerator is 1)

Wait — the expression simplifies to $\frac{1}{4}$, so numerator is 1.

Coloring instruction: “Numerator (purple)” → so 1 → purple

---

12. $s^2 \cdot s^{-5} \cdot s^{-4}$

Add exponents: 2 + (-5) + (-4) = -7

So $s^{-7} = \frac{1}{s^7}$

Exponent of the denominator: 7

→ Color pink

---

Now, compiling all solutions and colors:

1. Solution: 45 → purple
2. Solution: $6k^7g^5$ → coeff 6 (yellow), exp of g=5 (dark blue)
3. Solution: $\frac{1}{7^4}$ → exponent in denom = 4 → red
4. Solution: $\frac{5}{8h^2}$ → coeff in denom=8 (pink), exp of h=2 (light blue)
5. Solution: $\frac{5n}{7s^3}$ → var in num = n → purple
6. Solution: $\frac{5}{9w^4y^4}$ → sum: 9 + 4 = 13 → dark blue
7. Solution: $\frac{1}{6^4}$ → exponent in denom = 4 → red
8. Solution: $\frac{1}{n^{14}}$ → power of base = 14 → light blue
9. Solution: $8b^4s^6$ → sum exp b+s = 4+6=10 → yellow
10. Solution: $\frac{6h^5}{5d^8}$ → var in denom = d → pink
11. Solution: $\frac{1}{4}$ → numerator = 1 → purple
12. Solution: $\frac{1}{s^7}$ → exponent of denom = 7 → pink

Final Answer:
All expressions simplified with positive exponents, and coloring instructions applied as above.