Exponent Rules Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem one by one using exponent rules. We’ll simplify step by step.

---

Problem 1:
$\left(\frac{3xy^3}{5x^2y}\right)^4$

First, simplify inside the parentheses:

- $x / x^2 = x^{1-2} = x^{-1}$
- $y^3 / y = y^{3-1} = y^2$
- Constants: $3/5$ stays as is

So inside becomes: $\frac{3}{5} \cdot x^{-1} \cdot y^2 = \frac{3y^2}{5x}$

Now raise to the 4th power:

$(\frac{3y^2}{5x})^4 = \frac{3^4 (y^2)^4}{5^4 x^4} = \frac{81 y^8}{625 x^4}$

✔ Final Answer for #1: $\boxed{\frac{81y^8}{625x^4}}$

---

Problem 2:
$(3x^3y^5)(9x^4y^2)$

Multiply coefficients: $3 \cdot 9 = 27$

Add exponents for same bases:

- $x^3 \cdot x^4 = x^{3+4} = x^7$
- $y^5 \cdot y^2 = y^{5+2} = y^7$

✔ Final Answer for #2: $\boxed{27x^7y^7}$

---

Problem 3:
$(6x^3y^7z^4)^{-3}$

Apply the -3 exponent to each part:

- $6^{-3} = \frac{1}{6^3} = \frac{1}{216}$
- $(x^3)^{-3} = x^{-9}$
- $(y^7)^{-3} = y^{-21}$
- $(z^4)^{-3} = z^{-12}$

So we get: $\frac{1}{216} x^{-9} y^{-21} z^{-12}$

To write with positive exponents, move variables to denominator:

✔ Final Answer for #3: $\boxed{\frac{1}{216x^9y^{21}z^{12}}}$

---

Problem 4:
$\frac{7x^5y^7}{14x^3y^6}$

Simplify constants: $7/14 = 1/2$

Subtract exponents:

- $x^{5-3} = x^2$
- $y^{7-6} = y^1 = y$

✔ Final Answer for #4: $\boxed{\frac{x^2y}{2}}$

---

Problem 5:
$4x^{-2}y^{-3}$

Negative exponents mean “move to denominator”:

$x^{-2} = \frac{1}{x^2}, y^{-3} = \frac{1}{y^3}$

So: $4 \cdot \frac{1}{x^2} \cdot \frac{1}{y^3} = \frac{4}{x^2y^3}$

✔ Final Answer for #5: $\boxed{\frac{4}{x^2y^3}}$

---

Problem 6:
$(2x^{-4}y^6)(xy^9)$

Multiply coefficients: $2 \cdot 1 = 2$

Add exponents:

- $x^{-4} \cdot x^1 = x^{-4+1} = x^{-3}$
- $y^6 \cdot y^9 = y^{15}$

So: $2x^{-3}y^{15}$ → move $x^{-3}$ to denominator

✔ Final Answer for #6: $\boxed{\frac{2y^{15}}{x^3}}$

---

Problem 7:
$\left(\frac{2xyx}{3x^5y}\right)^0$

Anything (except zero) raised to the 0 power is 1.

Even if the inside looks messy — as long as it’s not zero, the whole thing is 1.

Check: numerator has $2xyx = 2x^2y$, denominator $3x^5y$. Not zero → so answer is 1.

✔ Final Answer for #7: $\boxed{1}$

---

Problem 8:
$\frac{6y^{12}}{2x^{-3}y^4}$

Simplify constants: $6/2 = 3$

Handle exponents:

- $y^{12}/y^4 = y^{8}$
- $x^{-3}$ in denominator → move to numerator as $x^3$

So: $3 \cdot x^3 \cdot y^8$

✔ Final Answer for #8: $\boxed{3x^3y^8}$

---

Final Answer:
1. $\boxed{\frac{81y^8}{625x^4}}$
2. $\boxed{27x^7y^7}$
3. $\boxed{\frac{1}{216x^9y^{21}z^{12}}}$
4. $\boxed{\frac{x^2y}{2}}$
5. $\boxed{\frac{4}{x^2y^3}}$
6. $\boxed{\frac{2y^{15}}{x^3}}$
7. $\boxed{1}$
8. $\boxed{3x^3y^8}$