Let's solve each expression step by step using the
Power Rule of exponents.
---
🔷 Power Rule:
When you have a power raised to another power, you
multiply the exponents:
$$
(a^m)^n = a^{m \cdot n}
$$
Also, remember:
- $ a^0 = 1 $ for any $ a \neq 0 $
- When multiplying variables with exponents: $ (ab)^n = a^n b^n $
---
✔ Left Column:
1. $ (x^2)^2 = x^{2 \cdot 2} = x^4 $
2. $ (x^2)^3 = x^{2 \cdot 3} = x^6 $
3. $ (x^2)^6 = x^{2 \cdot 6} = x^{12} $
4. $ (y^3)^2 = y^{3 \cdot 2} = y^6 $
5. $ (x^4)^0 = x^{4 \cdot 0} = x^0 = 1 $
---
✔ Right Column:
6. $ (x^2 y^2)^3 = (x^2)^3 \cdot (y^2)^3 = x^{6} y^{6} $
7. $ (x^2 y^4)^3 = (x^2)^3 \cdot (y^4)^3 = x^{6} y^{12} $
8. $ (x y^4)^3 = x^3 \cdot (y^4)^3 = x^3 y^{12} $
9. $ (x y^6)^4 = x^4 \cdot (y^6)^4 = x^4 y^{24} $
10. $ (x^0)^5 = (1)^5 = 1 $
(Because $ x^0 = 1 $, and $ 1^5 = 1 $)
---
✔ Final Answers:
| Expression | Simplified |
|----------|------------|
| $ (x^2)^2 $ | $ x^4 $ |
| $ (x^2)^3 $ | $ x^6 $ |
| $ (x^2)^6 $ | $ x^{12} $ |
| $ (y^3)^2 $ | $ y^6 $ |
| $ (x^4)^0 $ | $ 1 $ |
| $ (x^2 y^2)^3 $ | $ x^6 y^6 $ |
| $ (x^2 y^4)^3 $ | $ x^6 y^{12} $ |
| $ (x y^4)^3 $ | $ x^3 y^{12} $ |
| $ (x y^6)^4 $ | $ x^4 y^{24} $ |
| $ (x^0)^5 $ | $ 1 $ |
---
📝 Summary:
We applied the
power rule: $ (a^m)^n = a^{m \cdot n} $, and when dealing with products inside parentheses, we apply the exponent to each factor separately.
✔ All expressions are now simplified!
Parent Tip: Review the logic above to help your child master the concept of exponent rules worksheet 8th grade.
