Here are the step-by-step solutions for each problem using the laws of exponents.
1. $3^2 \times 3^4 \times 3^8$
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Rule: When multiplying terms with the same base, add the exponents ($a^m \times a^n = a^{m+n}$).
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Step: Add the powers: $2 + 4 + 8 = 14$.
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Result: The base is 3 and the new exponent is 14.
2. $6^{15} \div 6^{10}$
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Rule: When dividing terms with the same base, subtract the exponents ($a^m \div a^n = a^{m-n}$).
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Step: Subtract the powers: $15 - 10 = 5$.
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Result: The base is 6 and the new exponent is 5.
3. $a^3 \times a^2$
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Rule: Same base multiplication means adding exponents.
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Step: Add the powers: $3 + 2 = 5$.
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Result: The base is $a$ and the new exponent is 5.
4. $7^3 \times 7^2$
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Rule: Same base multiplication means adding exponents.
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Step: Add the powers: $3 + 2 = 5$.
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Result: The base is 7 and the new exponent is 5.
5. $2^5 \times 5^5$
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Rule: When bases are different but exponents are the same, you can combine the bases ($a^n \times b^n = (ab)^n$).
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Step: Multiply the bases: $2 \times 5 = 10$. Keep the exponent 5.
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Result: The base is 10 and the exponent is 5.
6. $a^4 \times b^4$
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Rule: Different bases, same exponent. Combine the bases.
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Step: Multiply the bases: $a \times b = ab$. Keep the exponent 4.
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Result: The base is $ab$ and the exponent is 4.
7. $(2^{20} \div 2^{15}) \times 2^3$
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Rule: Follow the order of operations (brackets first). Inside the brackets, subtract exponents. Then multiply by adding exponents.
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Step 1 (Brackets): $20 - 15 = 5$. So, $(2^{20} \div 2^{15})$ becomes $2^5$.
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Step 2 (Multiply): Now solve $2^5 \times 2^3$. Add the exponents: $5 + 3 = 8$.
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Result: The base is 2 and the new exponent is 8.
8. $8^8 \div 8^2$
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Rule: Same base division means subtracting exponents.
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Step: Subtract the powers: $8 - 2 = 6$.
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Result: The base is 8 and the new exponent is 6.
9. $3^4 \times 3^3 \times 3^6$
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Rule: Same base multiplication means adding all exponents together.
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Step: Add the powers: $4 + 3 + 6 = 13$.
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Result: The base is 3 and the new exponent is 13.
10. $6^{15} \times 6^{10}$
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Rule: Same base multiplication means adding exponents.
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Step: Add the powers: $15 + 10 = 25$.
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Result: The base is 6 and the new exponent is 25.
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Final Answer:
1. $3^{14}$
2. $6^5$
3. $a^5$
4. $7^5$
5. $10^5$
6. $(ab)^4$
7. $2^8$
8. $8^6$
9. $3^{13}$
10. $6^{25}$
Parent Tip: Review the logic above to help your child master the concept of power and exponents worksheet.
