To solve the problems in the worksheet, we need to use the
rule of multiplying exponents with the same base. The rule states:
$$
a^m \times a^n = a^{m+n}
$$
This means when you multiply two powers with the same base, you add their exponents.
Let's solve each problem step by step.
---
Problem 1:
$$
3^2 \times 3^4 \times 3^6 \times 3^{-5}
$$
- All terms have the same base, which is 3.
- Add the exponents: $2 + 4 + 6 - 5$.
- Calculation: $2 + 4 + 6 - 5 = 7$.
- Final answer: $3^7$.
$$
\boxed{3^7}
$$
---
Problem 2:
$$
2^2 \times 2^3 \times 2^4 \times 2^6
$$
- All terms have the same base, which is 2.
- Add the exponents: $2 + 3 + 4 + 6$.
- Calculation: $2 + 3 + 4 + 6 = 15$.
- Final answer: $2^{15}$.
$$
\boxed{2^{15}}
$$
---
Problem 3:
$$
4^3 \times 4^2 \times 4^4 \times 4^3 \times 4^6
$$
- All terms have the same base, which is 4.
- Add the exponents: $3 + 2 + 4 + 3 + 6$.
- Calculation: $3 + 2 + 4 + 3 + 6 = 18$.
- Final answer: $4^{18}$.
$$
\boxed{4^{18}}
$$
---
Problem 4:
$$
6^4 \times 6^2 \times 6^{11}
$$
- All terms have the same base, which is 6.
- Add the exponents: $4 + 2 + 11$.
- Calculation: $4 + 2 + 11 = 17$.
- Final answer: $6^{17}$.
$$
\boxed{6^{17}}
$$
---
Problem 5:
$$
a^5 \times a^2 \times a^3 \times a^4
$$
- All terms have the same base, which is $a$.
- Add the exponents: $5 + 2 + 3 + 4$.
- Calculation: $5 + 2 + 3 + 4 = 14$.
- Final answer: $a^{14}$.
$$
\boxed{a^{14}}
$$
---
Problem 6:
$$
3^2 \times 3^7 \times 3^2 \times 3^6 \times 3^{-4} \times 3^2 \times 3^5
$$
- All terms have the same base, which is 3.
- Add the exponents: $2 + 7 + 2 + 6 - 4 + 2 + 5$.
- Calculation: $2 + 7 + 2 + 6 - 4 + 2 + 5 = 20$.
- Final answer: $3^{20}$.
$$
\boxed{3^{20}}
$$
---
Problem 7:
$$
z^4 \times z^2 \times z^3 \times z^5 \times z^6
$$
- All terms have the same base, which is $z$.
- Add the exponents: $4 + 2 + 3 + 5 + 6$.
- Calculation: $4 + 2 + 3 + 5 + 6 = 20$.
- Final answer: $z^{20}$.
$$
\boxed{z^{20}}
$$
---
Problem 8:
$$
7^3 \times 7^2 \times 7^4 \times 7^{-2} \times 7^5
$$
- All terms have the same base, which is 7.
- Add the exponents: $3 + 2 + 4 - 2 + 5$.
- Calculation: $3 + 2 + 4 - 2 + 5 = 12$.
- Final answer: $7^{12}$.
$$
\boxed{7^{12}}
$$
---
Problem 9:
$$
2^6 \times 2^{-3} \times 2^5 \times 2^4
$$
- All terms have the same base, which is 2.
- Add the exponents: $6 - 3 + 5 + 4$.
- Calculation: $6 - 3 + 5 + 4 = 12$.
- Final answer: $2^{12}$.
$$
\boxed{2^{12}}
$$
---
Problem 10:
$$
6^2 \times 6^3 \times 6^4 \times 6^6 \times 6^4 \times 6^2
$$
- All terms have the same base, which is 6.
- Add the exponents: $2 + 3 + 4 + 6 + 4 + 2$.
- Calculation: $2 + 3 + 4 + 6 + 4 + 2 = 21$.
- Final answer: $6^{21}$.
$$
\boxed{6^{21}}
$$
---
Final Answers:
1. $3^7$
2. $2^{15}$
3. $4^{18}$
4. $6^{17}$
5. $a^{14}$
6. $3^{20}$
7. $z^{20}$
8. $7^{12}$
9. $2^{12}$
10. $6^{21}$
$$
\boxed{3^7, 2^{15}, 4^{18}, 6^{17}, a^{14}, 3^{20}, z^{20}, 7^{12}, 2^{12}, 6^{21}}
$$
Parent Tip: Review the logic above to help your child master the concept of multiplying exponents worksheet with answers.
