Exponential expressions worksheet focusing on multiplying and dividing, featuring sections for finding values, simplifying expressions, and an extension activity.

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Here are the solutions for each section of the worksheet:

Section A: Find the values.

1. $3^0$
* Any non-zero number raised to the power of 0 is 1.
* Answer: 1

2. $(-2)^3$
* Multiply -2 by itself three times: $(-2) \times (-2) \times (-2)$.
* $(-2) \times (-2) = 4$
* $4 \times (-2) = -8$
* Answer: -8

3. $(-1)^{20}$
* A negative number raised to an even power becomes positive.
* $(-1)$ multiplied by itself 20 times results in 1.
* Answer: 1

4. $7^3$
* Multiply 7 by itself three times: $7 \times 7 \times 7$.
* $7 \times 7 = 49$
* $49 \times 7 = 343$
* Answer: 343

5. $(-5)^3 + (-4)^2$
* Calculate $(-5)^3$: $(-5) \times (-5) \times (-5) = -125$.
* Calculate $(-4)^2$: $(-4) \times (-4) = 16$.
* Add them together: $-125 + 16 = -109$.
* Answer: -109

6. $(\frac{3}{4})^2$
* Square both the numerator and the denominator: $\frac{3^2}{4^2}$.
* $3^2 = 9$
* $4^2 = 16$
* Answer: $\frac{9}{16}$

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Section B: Simplify.

*(Rules: When multiplying with same base, add exponents. When dividing, subtract exponents.)*

1. $5^2 \times 5^4$
* Add exponents: $2 + 4 = 6$.
* Answer: $5^6$

2. $9^{11} \times 9^2$
* Add exponents: $11 + 2 = 13$.
* Answer: $9^{13}$

3. $2^5 \times 2$
* Remember that $2$ is the same as $2^1$.
* Add exponents: $5 + 1 = 6$.
* Answer: $2^6$

4. $8^5 \times 8^0$
* Add exponents: $5 + 0 = 5$.
* Answer: $8^5$

5. $7^5 \div 7^3$
* Subtract exponents: $5 - 3 = 2$.
* Answer: $7^2$

6. $12^{10} \div 12^6$
* Subtract exponents: $10 - 6 = 4$.
* Answer: $12^4$

7. $4^{21} \div 4$
* Remember that $4$ is the same as $4^1$.
* Subtract exponents: $21 - 1 = 20$.
* Answer: $4^{20}$

8. $5^3 \div 5^3$
* Subtract exponents: $3 - 3 = 0$.
* Answer: $5^0$ (or 1)

9. $2^{-3} \times 2^8$
* Add exponents: $-3 + 8 = 5$.
* Answer: $2^5$

10. $6^4 \div 6^{12}$
* Subtract exponents: $4 - 12 = -8$.
* Answer: $6^{-8}$

11. $15^2 \times 15^{-9}$
* Add exponents: $2 + (-9) = -7$.
* Answer: $15^{-7}$

12. $7^{10} \div 7^{14} \times 7$
* Order of operations: Divide first, then multiply.
* $7^{10} \div 7^{14} = 7^{10-14} = 7^{-4}$.
* Then multiply by 7 (which is $7^1$): $7^{-4} \times 7^1 = 7^{-4+1} = 7^{-3}$.
* Answer: $7^{-3}$

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Section C: Simplify.

1. $\frac{2^{10}}{2^4}$
* Subtract exponents: $10 - 4 = 6$.
* Answer: $2^6$

2. $\frac{8^{-1}}{8^2}$
* Subtract exponents: $-1 - 2 = -3$.
* Answer: $8^{-3}$

3. $\frac{10^{-12}}{10^7}$
* Subtract exponents: $-12 - 7 = -19$.
* Answer: $10^{-19}$

4. $\frac{3^{-6}}{3^{-1}}$
* Subtract exponents: $-6 - (-1) = -6 + 1 = -5$.
* Answer: $3^{-5}$

5. $\frac{3^3 \times 3^5}{3^2}$
* Top part (numerator): Add exponents $3 + 5 = 8$, so $3^8$.
* Now divide by bottom: $8 - 2 = 6$.
* Answer: $3^6$

6. $\frac{6^{-2} \times 6^4}{6^{-8}}$
* Top part: $-2 + 4 = 2$, so $6^2$.
* Now divide by bottom: $2 - (-8) = 2 + 8 = 10$.
* Answer: $6^{10}$

7. $\frac{7^{-8} \times 7^{-7}}{7^{-4}}$
* Top part: $-8 + (-7) = -15$, so $7^{-15}$.
* Now divide by bottom: $-15 - (-4) = -15 + 4 = -11$.
* Answer: $7^{-11}$

8. $\frac{14^{-3}}{14^{-11} \times 14^{-19}}$
* Bottom part: $-11 + (-19) = -30$, so $14^{-30}$.
* Now divide top by bottom: $-3 - (-30) = -3 + 30 = 27$.
* Answer: $14^{27}$

9. $\frac{4^8 \times 4^5}{4^2 \times 4^4}$
* Top part: $8 + 5 = 13$, so $4^{13}$.
* Bottom part: $2 + 4 = 6$, so $4^6$.
* Divide: $13 - 6 = 7$.
* Answer: $4^7$

10. $\frac{6^{-7} \times 6^3}{6^{-1} \times 6}$
* Top part: $-7 + 3 = -4$, so $6^{-4}$.
* Bottom part: $-1 + 1 = 0$, so $6^0$ (which is 1).
* Divide: $-4 - 0 = -4$.
* Answer: $6^{-4}$

11. $\frac{20^{-17} \times 20^9}{20^4 \times 20^{-11}}$
* Top part: $-17 + 9 = -8$, so $20^{-8}$.
* Bottom part: $4 + (-11) = -7$, so $20^{-7}$.
* Divide: $-8 - (-7) = -8 + 7 = -1$.
* Answer: $20^{-1}$

12. $\frac{5e^8 \times 4e^5}{2e^2 \times 10e^4}$
* Simplify numbers first: Top is $5 \times 4 = 20$. Bottom is $2 \times 10 = 20$.
* $20 \div 20 = 1$.
* Simplify variables ($e$): Top exponents $8 + 5 = 13$. Bottom exponents $2 + 4 = 6$.
* Divide exponents: $13 - 6 = 7$.
* Answer: $e^7$

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Extension: Investigate the following

Column 1:
* $(2^1)^2 = 2^1 \times 2^1 = 2^2$ (Given)
* $(2^2)^2 = 2^2 \times 2^2 = 2^4$
* $(2^3)^2 = 2^3 \times 2^3 = 2^6$
* $(2^4)^2 = 2^4 \times 2^4 = 2^8$
* $(2^5)^2 = 2^5 \times 2^5 = 2^{10}$

Column 2:
* $(5^1)^3 = 5^1 \times 5^1 \times 5^1 = 5^3$ (Given)
* $(5^2)^3 = 5^2 \times 5^2 \times 5^2 = 5^6$
* $(5^3)^3 = 5^3 \times 5^3 \times 5^3 = 5^9$
* $(5^4)^3 = 5^4 \times 5^4 \times 5^4 = 5^{12}$
* $(5^5)^3 = 5^5 \times 5^5 \times 5^5 = 5^{15}$

Column 3 (Use your results to simplify):
* $(4^2)^4 = 4^{2 \times 4} = 4^8$
* $(6^3)^8 = 6^{3 \times 8} = 6^{24}$
* $(9^{-5})^6 = 9^{-5 \times 6} = 9^{-30}$
* $(11^{-2})^{-7} = 11^{-2 \times -7} = 11^{14}$