Simplifying Expressions with Exponents Worksheets - Free Printable

To solve the given problems involving simplifying exponential expressions, we will use the rules of exponents. Here are the key rules we will apply:

1. Product Rule: $ a^m \cdot a^n = a^{m+n} $
2. Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
3. Power Rule: $ (a^m)^n = a^{m \cdot n} $
4. Negative Exponent Rule: $ a^{-n} = \frac{1}{a^n} $

Let's solve each problem step by step.

---

Problem 1:

$$
\frac{14a^{-10}}{7a^{-8}}
$$

#### Step 1: Simplify the coefficients.
$$
\frac{14}{7} = 2
$$

#### Step 2: Apply the quotient rule for exponents.
$$
\frac{a^{-10}}{a^{-8}} = a^{-10 - (-8)} = a^{-10 + 8} = a^{-2}
$$

#### Step 3: Combine the results.
$$
\frac{14a^{-10}}{7a^{-8}} = 2a^{-2}
$$

#### Final Answer:
$$
\boxed{2a^{-2}}
$$

---

Problem 2:

$$
(-19t^6)(-5t^7)
$$

#### Step 1: Multiply the coefficients.
$$
(-19)(-5) = 95
$$

#### Step 2: Apply the product rule for exponents.
$$
t^6 \cdot t^7 = t^{6+7} = t^{13}
$$

#### Step 3: Combine the results.
$$
(-19t^6)(-5t^7) = 95t^{13}
$$

#### Final Answer:
$$
\boxed{95t^{13}}
$$

---

Problem 3:

$$
(-3m^{-7})(9m^{14})
$$

#### Step 1: Multiply the coefficients.
$$
(-3)(9) = -27
$$

#### Step 2: Apply the product rule for exponents.
$$
m^{-7} \cdot m^{14} = m^{-7 + 14} = m^7
$$

#### Step 3: Combine the results.
$$
(-3m^{-7})(9m^{14}) = -27m^7
$$

#### Final Answer:
$$
\boxed{-27m^7}
$$

---

Problem 4:

$$
\frac{2s^8}{18s^{-6}}
$$

#### Step 1: Simplify the coefficients.
$$
\frac{2}{18} = \frac{1}{9}
$$

#### Step 2: Apply the quotient rule for exponents.
$$
\frac{s^8}{s^{-6}} = s^{8 - (-6)} = s^{8 + 6} = s^{14}
$$

#### Step 3: Combine the results.
$$
\frac{2s^8}{18s^{-6}} = \frac{1}{9}s^{14}
$$

#### Final Answer:
$$
\boxed{\frac{s^{14}}{9}}
$$

---

Problem 5:

$$
\frac{6r^8}{8r^7}
$$

#### Step 1: Simplify the coefficients.
$$
\frac{6}{8} = \frac{3}{4}
$$

#### Step 2: Apply the quotient rule for exponents.
$$
\frac{r^8}{r^7} = r^{8-7} = r^1 = r
$$

#### Step 3: Combine the results.
$$
\frac{6r^8}{8r^7} = \frac{3}{4}r
$$

#### Final Answer:
$$
\boxed{\frac{3r}{4}}
$$

---

Problem 6:

$$
(5z^9)(11z^{-2})
$$

#### Step 1: Multiply the coefficients.
$$
(5)(11) = 55
$$

#### Step 2: Apply the product rule for exponents.
$$
z^9 \cdot z^{-2} = z^{9 + (-2)} = z^{9 - 2} = z^7
$$

#### Step 3: Combine the results.
$$
(5z^9)(11z^{-2}) = 55z^7
$$

#### Final Answer:
$$
\boxed{55z^7}
$$

---

Problem 7:

$$
(15v^{-9})(-2v^{-3})
$$

#### Step 1: Multiply the coefficients.
$$
(15)(-2) = -30
$$

#### Step 2: Apply the product rule for exponents.
$$
v^{-9} \cdot v^{-3} = v^{-9 + (-3)} = v^{-12}
$$

#### Step 3: Combine the results.
$$
(15v^{-9})(-2v^{-3}) = -30v^{-12}
$$

#### Final Answer:
$$
\boxed{-30v^{-12}}
$$

---

Problem 8:

$$
\frac{20u^{-4}}{5u^{12}}
$$

#### Step 1: Simplify the coefficients.
$$
\frac{20}{5} = 4
$$

#### Step 2: Apply the quotient rule for exponents.
$$
\frac{u^{-4}}{u^{12}} = u^{-4 - 12} = u^{-16}
$$

#### Step 3: Combine the results.
$$
\frac{20u^{-4}}{5u^{12}} = 4u^{-16}
$$

#### Final Answer:
$$
\boxed{4u^{-16}}
$$

---

Final Answers:

1. $\boxed{2a^{-2}}$
2. $\boxed{95t^{13}}$
3. $\boxed{-27m^7}$
4. $\boxed{\frac{s^{14}}{9}}$
5. $\boxed{\frac{3r}{4}}$
6. $\boxed{55z^7}$
7. $\boxed{-30v^{-12}}$
8. $\boxed{4u^{-16}}$