Let’s go through each problem step by step using exponent rules.
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Exponent Rules Section:
1. $ a^n \cdot a^m = a^{n+m} $ → When multiplying same base, add exponents.
2. $ a^n b^n = (ab)^n $ → When multiplying different bases with same exponent, combine bases under one exponent.
3. $ \frac{a^n}{a^m} = a^{n-m} $ → When dividing same base, subtract exponents.
4. $ a^n \div b^n = \left(\frac{a}{b}\right)^n $ → When dividing different bases with same exponent, write as fraction raised to that exponent.
5. $ (a^n)^m = a^{n \cdot m} $ → When raising a power to another power, multiply exponents.
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Simplify Problems:
1. $ a^2 a^5 = a^{2+5} = a^7 $
2. $ \frac{5^6}{5^4} = 5^{6-4} = 5^2 = 25 $
3. $ (2x)^2 = 2^2 \cdot x^2 = 4x^2 $
4. $ (2^3 x^7)^0 = 1 $ → Anything (except 0) to the power of 0 is 1.
5. $ \left(\frac{3}{x}\right)^4 = \frac{3^4}{x^4} = \frac{81}{x^4} $
6. $ (x^2)^3 = x^{2 \cdot 3} = x^6 $
7. $ 2^2 \cdot 2^4 \cdot 2^{20} = 2^{2+4+20} = 2^{26} $
8. $ (3x^2)^2 = 3^2 \cdot (x^2)^2 = 9x^4 $
9. $ \left(\frac{3^{12}}{3^8}\right)^2 = (3^{12-8})^2 = (3^4)^2 = 3^{8} = 6561 $
10. $ \frac{a^5 b^{10}}{a^3 b^8} = a^{5-3} b^{10-8} = a^2 b^2 $
11. $ w^6 \div w^4 = w^{6-4} = w^2 $
12. $ 3^5 \times 4^5 = (3 \cdot 4)^5 = 12^5 = 248832 $
13. $ 2^4 \times 2^5 \times 2^6 \div 2^{15} = 2^{4+5+6-15} = 2^{0} = 1 $
14. $ ((3^4)^5)^2 = 3^{4 \cdot 5 \cdot 2} = 3^{40} $
15. $ \frac{4^6 \times 3^6}{6^6} = \left( \frac{4 \cdot 3}{6} \right)^6 = \left( \frac{12}{6} \right)^6 = 2^6 = 64 $
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Final Answer:
1. $ a^7 $
2. $ 25 $
3. $ 4x^2 $
4. $ 1 $
5. $ \frac{81}{x^4} $
6. $ x^6 $
7. $ 2^{26} $
8. $ 9x^4 $
9. $ 6561 $
10. $ a^2 b^2 $
11. $ w^2 $
12. $ 248832 $
13. $ 1 $
14. $ 3^{40} $
15. $ 64 $
Parent Tip: Review the logic above to help your child master the concept of algebra 2 exponents worksheet.
