Let’s solve each problem step by step using the laws of exponents.
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1. \( a^3 \cdot a^5 = a^{3+5} = a^8 \)
→ Add exponents when multiplying same base.
2. \( a^3b \cdot ab^6 = a^{3+1} b^{1+6} = a^4 b^7 \)
→ Multiply coefficients (here, both are 1), add exponents for same variables.
3. \( (2b^5)^3 = 2^3 \cdot (b^5)^3 = 8b^{15} \)
→ Raise coefficient and variable to power: \( (ab)^n = a^n b^n \), and \( (b^m)^n = b^{m\cdot n} \)
4. \( (a^4 b^2)^3 = a^{4\cdot3} b^{2\cdot3} = a^{12} b^6 \)
→ Apply exponent to each part inside parentheses.
5. \( (a^2)^3 (a^2)^2 = a^{6} \cdot a^{4} = a^{6+4} = a^{10} \)
→ First simplify powers: \( (a^m)^n = a^{m\cdot n} \), then multiply by adding exponents.
6. \( (3b^2)(2b) = 3\cdot2 \cdot b^{2+1} = 6b^3 \)
→ Multiply numbers, add exponents for same variable.
7. \( \frac{a^7}{a^3} = a^{7-3} = a^4 \)
→ Subtract exponents when dividing same base.
8. \( \left( \frac{a^3 b^5}{ab} \right)^2 = \left( a^{3-1} b^{5-1} \right)^2 = (a^2 b^4)^2 = a^{4} b^{8} \)
→ Simplify inside first by subtracting exponents, then square everything.
9. \( \frac{(4a^4)^2}{(2a)^3} = \frac{16a^8}{8a^3} = \frac{16}{8} \cdot a^{8-3} = 2a^5 \)
→ Expand numerator and denominator separately, then divide coefficients and subtract exponents.
10. \( (5a^2bc^4)^0 = 1 \)
→ Anything (except zero) raised to power 0 is 1.
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Final Answer:
1. \( a^8 \)
2. \( a^4 b^7 \)
3. \( 8b^{15} \)
4. \( a^{12} b^6 \)
5. \( a^{10} \)
6. \( 6b^3 \)
7. \( a^4 \)
8. \( a^4 b^8 \)
9. \( 2a^5 \)
10. \( 1 \)
Parent Tip: Review the logic above to help your child master the concept of laws of exponents worksheet answers.
