Worksheet for practicing exponent rules, including examples and exercises on simplifying expressions with exponents.

Here are the step-by-step solutions for the problems on the worksheet. I have grouped them by the rules used to make it easier to follow.

Product Rule Problems

*Rule: When multiplying terms with the same base, add the exponents ($x^a \cdot x^b = x^{a+b}$). Also, multiply the coefficients (numbers).*

1) $a \cdot a^2 \cdot a^3$
* Think of $a$ as $a^1$.
* Add exponents: $1 + 2 + 3 = 6$.
* Answer: $a^6$

2) $(2a^2b)(4ab^3)$
* Multiply numbers: $2 \cdot 4 = 8$.
* Add $a$ exponents: $2 + 1 = 3 \rightarrow a^3$.
* Add $b$ exponents: $1 + 3 = 4 \rightarrow b^4$.
* Answer: $8a^3b^4$

3) $(6x^2)(-3x^5)$
* Multiply numbers: $6 \cdot -3 = -18$.
* Add $x$ exponents: $2 + 5 = 7$.
* Answer: $-18x^7$

4) $b^3 \cdot b^4 \cdot b^7 \cdot b$
* Think of the last $b$ as $b^1$.
* Add exponents: $3 + 4 + 7 + 1 = 15$.
* Answer: $b^{15}$

5) $(3x^4)(3x^6)(-3x^2)$
* Multiply numbers: $3 \cdot 3 \cdot -3 = -27$.
* Add $x$ exponents: $4 + 6 + 2 = 12$.
* Answer: $-27x^{12}$

16) $x^2 \cdot x^7$
* Add exponents: $2 + 7 = 9$.
* Answer: $x^9$

19) $2x^3 + 7x^3$
* *Note:* This is addition, not multiplication. You just add the coefficients because the variables are identical.
* $2 + 7 = 9$.
* Answer: $9x^3$

---

Power Rule Problems

*Rule: When raising a power to another power, multiply the exponents $(x^a)^b = x^{a \cdot b}$. Apply the outer exponent to every part inside the parentheses.*

6) $(2x^2y^3)^2$
* Square the number: $2^2 = 4$.
* Multiply $x$ exponents: $2 \cdot 2 = 4$.
* Multiply $y$ exponents: $3 \cdot 2 = 6$.
* Answer: $4x^4y^6$

7) $(5x^2y^4)^3$
* Cube the number: $5^3 = 125$.
* Multiply $x$ exponents: $2 \cdot 3 = 6$.
* Multiply $y$ exponents: $4 \cdot 3 = 12$.
* Answer: $125x^6y^{12}$

8) $(6x^3y^6)^4$
* Raise number to 4th power: $6^4 = 1296$.
* Multiply $x$ exponents: $3 \cdot 4 = 12$.
* Multiply $y$ exponents: $6 \cdot 4 = 24$.
* Answer: $1296x^{12}y^{24}$

9) $(4x^3y^3)^3$
* Cube the number: $4^3 = 64$.
* Multiply $x$ exponents: $3 \cdot 3 = 9$.
* Multiply $y$ exponents: $3 \cdot 3 = 9$.
* Answer: $64x^9y^9$

10) $(7xy)^2$
* Square the number: $7^2 = 49$.
* Square $x$: $x^2$.
* Square $y$: $y^2$.
* Answer: $49x^2y^2$

17) $(x^2)^7$
* Multiply exponents: $2 \cdot 7 = 14$.
* Answer: $x^{14}$

18) $(-2x^4)^5$
* Raise number to 5th power: $(-2)^5 = -32$ (negative stays negative because the power is odd).
* Multiply $x$ exponents: $4 \cdot 5 = 20$.
* Answer: $-32x^{20}$

29) $(3st^2)^3$
* Cube the number: $3^3 = 27$.
* Cube $s$: $s^3$.
* Multiply $t$ exponents: $2 \cdot 3 = 6$.
* Answer: $27s^3t^6$

30) $\left(\frac{3m^2n^{-7}}{m}\right)^3$
* First simplify inside: $\frac{m^2}{m^1} = m^1$. So inside is $3m^1n^{-7}$.
* Now apply the cube: $3^3 = 27$, $m^{1\cdot3}=m^3$, $n^{-7\cdot3}=n^{-21}$.
* Move negative exponent to denominator: $\frac{1}{n^{21}}$.
* Answer: $\frac{27m^3}{n^{21}}$

---

Quotient Rule Problems

*Rule: When dividing terms with the same base, subtract the bottom exponent from the top exponent ($\frac{x^a}{x^b} = x^{a-b}$). Simplify coefficients first.*

11) $\frac{x^3}{x}$
* Subtract exponents: $3 - 1 = 2$.
* Answer: $x^2$

12) $\frac{18c^3}{-3c^2}$
* Divide numbers: $18 / -3 = -6$.
* Subtract $c$ exponents: $3 - 2 = 1$.
* Answer: $-6c$

13) $\frac{9a^2b^5}{-3ab^2}$
* Divide numbers: $9 / -3 = -3$.
* Subtract $a$ exponents: $2 - 1 = 1$.
* Subtract $b$ exponents: $5 - 2 = 3$.
* Answer: $-3ab^3$

14) $\frac{-48c^2d^4}{-8cd}$
* Divide numbers: $-48 / -8 = 6$ (negatives cancel out).
* Subtract $c$ exponents: $2 - 1 = 1$.
* Subtract $d$ exponents: $4 - 1 = 3$.
* Answer: $6cd^3$

15) $\frac{22y^5z^2}{2yz^2}$
* Divide numbers: $22 / 2 = 11$.
* Subtract $y$ exponents: $5 - 1 = 4$.
* Subtract $z$ exponents: $2 - 2 = 0$. Anything to the power of 0 is 1, so $z$ disappears.
* Answer: $11y^4$

26) $\frac{2x^4}{-8x^4}$
* Divide numbers: $2 / -8 = -1/4$ (or $-0.25$).
* Subtract $x$ exponents: $4 - 4 = 0$. $x^0 = 1$, so $x$ disappears.
* Answer: $-\frac{1}{4}$

27) $\frac{3y^7}{3y^4}$
* Divide numbers: $3 / 3 = 1$.
* Subtract $y$ exponents: $7 - 4 = 3$.
* Answer: $y^3$

28) $6x^5 \cdot 3x^3 \cdot x^0$
* Remember $x^0 = 1$. So this is just $6x^5 \cdot 3x^3 \cdot 1$.
* Multiply numbers: $6 \cdot 3 = 18$.
* Add $x$ exponents: $5 + 3 = 8$.
* Answer: $18x^8$

---

Evaluation & Special Cases

20) $7^0$
* Any non-zero number raised to the power of 0 equals 1.
* Answer: $1$

21) $8x^0$
* Only the $x$ has the exponent 0. $x^0 = 1$.
* So, $8 \cdot 1 = 8$.
* Answer: $8$

22) $-3^4$
* Be careful! There are no parentheses around the $-3$. This means "the opposite of $3^4$".
* $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$.
* Apply the negative sign: $-81$.
* Answer: $-81$

23) $(-3)^4$
* The negative is inside the parentheses, so we multiply $-3$ four times.
* $(-3) \cdot (-3) \cdot (-3) \cdot (-3) = 81$.
* Answer: $81$

24) $6x^8y^2 - (2y^3)^4$
* First, simplify the power part: $(2y^3)^4 = 2^4 \cdot y^{3\cdot4} = 16y^{12}$.
* The expression becomes $6x^8y^2 - 16y^{12}$.
* These terms are not "like terms" (different variables and powers), so they cannot be combined further.
* Answer: $6x^8y^2 - 16y^{12}$

25) $(x + 2y)(x - 2y)$
* This is a "Difference of Squares" pattern: $(a+b)(a-b) = a^2 - b^2$.
* $x^2 - (2y)^2$.
* Square the second term: $(2y)^2 = 4y^2$.
* Answer: $x^2 - 4y^2$

Final Answer:
1) $a^6$
2) $8a^3b^4$
3) $-18x^7$
4) $b^{15}$
5) $-27x^{12}$
6) $4x^4y^6$
7) $125x^6y^{12}$
8) $1296x^{12}y^{24}$
9) $64x^9y^9$
10) $49x^2y^2$
11) $x^2$
12) $-6c$
13) $-3ab^3$
14) $6cd^3$
15) $11y^4$
16) $x^9$
17) $x^{14}$
18) $-32x^{20}$
19) $9x^3$
20) $1$
21) $8$
22) $-81$
23) $81$
24) $6x^8y^2 - 16y^{12}$
25) $x^2 - 4y^2$
26) $-\frac{1}{4}$
27) $y^3$
28) $18x^8$
29) $27s^3t^6$
30) $\frac{27m^3}{n^{21}}$