Solved first 10 questionExponent Rules Review Worksheet | Chegg.com - Free Printable

Let’s solve each problem step by step using the exponent rules given in the worksheet.

We’ll go one by one, applying:
- Product Rule: When multiplying same base → add exponents.
- Power Rule: When raising a power to another power → multiply exponents.
- Quotient Rule: When dividing same base → subtract exponents.
- Also remember: coefficients (numbers) are handled separately from variables.
- And: anything to the 0 power = 1.

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1) \( a \cdot a^2 \cdot a^3 \)
Same base “a”. Add exponents: \( 1 + 2 + 3 = 6 \)
→ \( a^6 \)

2) \( (2a^2b)(4ab^3) \)
Multiply coefficients: \( 2 \cdot 4 = 8 \)
For \( a \): \( a^2 \cdot a = a^{2+1} = a^3 \)
For \( b \): \( b \cdot b^3 = b^{1+3} = b^4 \)
→ \( 8a^3b^4 \)

3) \( (6x^2y)(-3x^5) \)
Coefficients: \( 6 \cdot (-3) = -18 \)
\( x^2 \cdot x^5 = x^{7} \)
Only one y → stays as y
→ \( -18x^7y \)

4) \( b^3 \cdot b^4 \cdot b^7 \cdot b \)
Add exponents: \( 3 + 4 + 7 + 1 = 15 \)
→ \( b^{15} \)

5) \( (3x^4)(3x^6)(-3x^2) \)
Coefficients: \( 3 \cdot 3 \cdot (-3) = -27 \)
Exponents for x: \( 4 + 6 + 2 = 12 \)
→ \( -27x^{12} \)

6) \( (2x^2y^3)^2 \)
Apply power rule to each part:
\( 2^2 = 4 \), \( (x^2)^2 = x^4 \), \( (y^3)^2 = y^6 \)
→ \( 4x^4y^6 \)

7) \( (5x^2y^4)^3 \)
\( 5^3 = 125 \), \( (x^2)^3 = x^6 \), \( (y^4)^3 = y^{12} \)
→ \( 125x^6y^{12} \)

8) \( (6x^4y^5)^3 \)
\( 6^3 = 216 \), \( x^{12} \), \( y^{15} \)
→ \( 216x^{12}y^{15} \)

9) \( (4x^3y^2)^3 \)
\( 4^3 = 64 \), \( x^9 \), \( y^6 \)
→ \( 64x^9y^6 \)

10) \( (7xy)^2 \)
\( 7^2 = 49 \), \( x^2 \), \( y^2 \)
→ \( 49x^2y^2 \)

11) \( \frac{x^3}{x} \)
Subtract exponents: \( 3 - 1 = 2 \)
→ \( x^2 \)

12) \( \frac{18c^3}{-3c^2} \)
Coefficients: \( 18 / (-3) = -6 \)
Exponents: \( c^{3-2} = c^1 = c \)
→ \( -6c \)

13) \( \frac{9a^2b^5}{-3ab^2} \)
Coefficients: \( 9 / (-3) = -3 \)
a: \( a^{2-1} = a \)
b: \( b^{5-2} = b^3 \)
→ \( -3ab^3 \)

14) \( \frac{-48c^2d^4}{-8cd} \)
Coefficients: \( -48 / -8 = 6 \)
c: \( c^{2-1} = c \)
d: \( d^{4-1} = d^3 \)
→ \( 6cd^3 \)

15) \( \frac{22y^5z^2}{2yz^2} \)
Coefficients: \( 22 / 2 = 11 \)
y: \( y^{5-1} = y^4 \)
z: \( z^{2-2} = z^0 = 1 \) → disappears
→ \( 11y^4 \)

16) \( x^2 \cdot x^7 \)
Add exponents: \( 2 + 7 = 9 \)
→ \( x^9 \)

17) \( (x^3)^2 \)
Multiply exponents: \( 3 \cdot 2 = 6 \)
→ \( x^6 \)

18) \( (-2x^4)^5 \)
First, \( (-2)^5 = -32 \)
Then \( (x^4)^5 = x^{20} \)
→ \( -32x^{20} \)

19) \( 2x^3 + 7x^3 \)
Like terms — just add coefficients: \( 2 + 7 = 9 \)
→ \( 9x^3 \)

20) \( 7^0 \)
Anything to 0 power = 1
→ \( 1 \)

21) \( 8x^0 \)
\( x^0 = 1 \), so \( 8 \cdot 1 = 8 \)
→ \( 8 \)

22) \( -3^4 \)
Careful! This is NOT \( (-3)^4 \). It means \( -(3^4) \)
\( 3^4 = 81 \), so answer is \( -81 \)
→ \( -81 \)

23) \( (-3)^4 \)
Negative raised to even power → positive
\( (-3)^4 = 81 \)
→ \( 81 \)

24) \( 6x^8y^2 - (2y^3)^4 \)
First simplify \( (2y^3)^4 = 2^4 \cdot y^{12} = 16y^{12} \)
So expression becomes: \( 6x^8y^2 - 16y^{12} \)
No like terms → leave as is
→ \( 6x^8y^2 - 16y^{12} \)

25) \( (x + 2y)(x - 2y) \)
This is difference of squares: \( (a+b)(a-b) = a^2 - b^2 \)
Here, \( a = x \), \( b = 2y \)
→ \( x^2 - (2y)^2 = x^2 - 4y^2 \)

26) \( \frac{2x^4}{-8x^8} \)
Coefficients: \( 2 / -8 = -\frac{1}{4} \)
Exponents: \( x^{4-8} = x^{-4} = \frac{1}{x^4} \)
So overall: \( -\frac{1}{4x^4} \)
→ \( -\frac{1}{4x^4} \)

27) \( \frac{3y^7}{x^2y^4} \)
y: \( y^{7-4} = y^3 \)
x stays in denominator
→ \( \frac{3y^3}{x^2} \)

28) \( 6x^2 \cdot 3x^5 \cdot x^0 \)
Coefficients: \( 6 \cdot 3 = 18 \)
x: \( x^{2+5+0} = x^7 \)
→ \( 18x^7 \)

29) \( (3st^2)^3 \)
\( 3^3 = 27 \), \( s^3 \), \( (t^2)^3 = t^6 \)
→ \( 27s^3t^6 \)

30) \( \left( \frac{3m^2n^7}{m} \right)^3 \)
First simplify inside:
\( \frac{3m^2n^7}{m} = 3m^{2-1}n^7 = 3mn^7 \)
Now raise to 3rd power:
\( (3)^3 = 27 \), \( m^3 \), \( (n^7)^3 = n^{21} \)
→ \( 27m^3n^{21} \)

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Final Answer:

1) \( a^6 \)
2) \( 8a^3b^4 \)
3) \( -18x^7y \)
4) \( b^{15} \)
5) \( -27x^{12} \)
6) \( 4x^4y^6 \)
7) \( 125x^6y^{12} \)
8) \( 216x^{12}y^{15} \)
9) \( 64x^9y^6 \)
10) \( 49x^2y^2 \)
11) \( x^2 \)
12) \( -6c \)
13) \( -3ab^3 \)
14) \( 6cd^3 \)
15) \( 11y^4 \)
16) \( x^9 \)
17) \( x^6 \)
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) \( -\frac{1}{4x^4} \)
27) \( \frac{3y^3}{x^2} \)
28) \( 18x^7 \)
29) \( 27s^3t^6 \)
30) \( 27m^3n^{21} \)