Power Rule of Exponents Worksheets (printable, online, answers ... - Free Printable

To solve the given problems, we will use the Power Rule of exponents, which states:

\[
(a^m)^n = a^{m \cdot n}
\]

Additionally, we will use other exponent rules as needed, such as:
- \((ab)^n = a^n b^n\)
- \(a^0 = 1\) (for any non-zero \(a\))

Let's solve each problem step by step.

---

Left Column

1. \((x^2)^2\)

Using the Power Rule:
\[
(x^2)^2 = x^{2 \cdot 2} = x^4
\]

2. \((x^2)^3\)

Using the Power Rule:
\[
(x^2)^3 = x^{2 \cdot 3} = x^6
\]

3. \((x^2)^6\)

Using the Power Rule:
\[
(x^2)^6 = x^{2 \cdot 6} = x^{12}
\]

4. \((y^3)^2\)

Using the Power Rule:
\[
(y^3)^2 = y^{3 \cdot 2} = y^6
\]

5. \((x^4)^0\)

Using the rule \(a^0 = 1\):
\[
(x^4)^0 = 1
\]

---

Right Column

1. \((x^2 y^2)^3\)

Using the rule \((ab)^n = a^n b^n\):
\[
(x^2 y^2)^3 = (x^2)^3 \cdot (y^2)^3
\]
Now apply the Power Rule to each term:
\[
(x^2)^3 = x^{2 \cdot 3} = x^6 \quad \text{and} \quad (y^2)^3 = y^{2 \cdot 3} = y^6
\]
Therefore:
\[
(x^2 y^2)^3 = x^6 y^6
\]

2. \((x^2 y^4)^5\)

Using the rule \((ab)^n = a^n b^n\):
\[
(x^2 y^4)^5 = (x^2)^5 \cdot (y^4)^5
\]
Now apply the Power Rule to each term:
\[
(x^2)^5 = x^{2 \cdot 5} = x^{10} \quad \text{and} \quad (y^4)^5 = y^{4 \cdot 5} = y^{20}
\]
Therefore:
\[
(x^2 y^4)^5 = x^{10} y^{20}
\]

3. \((xy^4)^3\)

Using the rule \((ab)^n = a^n b^n\):
\[
(xy^4)^3 = x^3 \cdot (y^4)^3
\]
Now apply the Power Rule to the second term:
\[
(y^4)^3 = y^{4 \cdot 3} = y^{12}
\]
Therefore:
\[
(xy^4)^3 = x^3 y^{12}
\]

4. \((xy^6)^4\)

Using the rule \((ab)^n = a^n b^n\):
\[
(xy^6)^4 = x^4 \cdot (y^6)^4
\]
Now apply the Power Rule to the second term:
\[
(y^6)^4 = y^{6 \cdot 4} = y^{24}
\]
Therefore:
\[
(xy^6)^4 = x^4 y^{24}
\]

5. \((x^0)^5\)

Using the rule \(a^0 = 1\):
\[
x^0 = 1
\]
Therefore:
\[
(x^0)^5 = 1^5 = 1
\]

---

Final Answers

Combining all the results, we have:

\[
\begin{aligned}
&\text{Left Column:} \\
&\quad (x^2)^2 = x^4 \\
&\quad (x^2)^3 = x^6 \\
&\quad (x^2)^6 = x^{12} \\
&\quad (y^3)^2 = y^6 \\
&\quad (x^4)^0 = 1 \\
\\
&\text{Right Column:} \\
&\quad (x^2 y^2)^3 = x^6 y^6 \\
&\quad (x^2 y^4)^5 = x^{10} y^{20} \\
&\quad (xy^4)^3 = x^3 y^{12} \\
&\quad (xy^6)^4 = x^4 y^{24} \\
&\quad (x^0)^5 = 1 \\
\end{aligned}
\]

\boxed{
\begin{aligned}
&x^4, \quad x^6, \quad x^{12}, \quad y^6, \quad 1 \\
&x^6 y^6, \quad x^{10} y^{20}, \quad x^3 y^{12}, \quad x^4 y^{24}, \quad 1
\end{aligned}
}