Indices Method A98575

1. **Evaluate (a) $5^3 \div (5^2 \times 5^{-3})$** 1. The problem asks us to divide $5^3$ by the product $5^2 \times 5^{-3}$. 2. Recall the index laws: when multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$. 3. So, $5^2 \times 5^{-3} = 5^{2 + (-3)} = 5^{-1}$. 4. Now the expression is $5^3 \div 5^{-1}$. 5. When dividing powers with the same base, subtract the exponents: $a^m \div a^n = a^{m-n}$. 6. So, $5^3 \div 5^{-1} = 5^{3 - (-1)} = 5^{3 + 1} = 5^4$. 7. The final answer is $5^4 = 625$. 2. **Evaluate (b) $\left(\frac{3}{2}\right)^2 \times \left(\frac{2}{3}\right)^{-2}$** 1. First, recall that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. 2. Also, $x^{-n} = \frac{1}{x^n}$. 3. Calculate $\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}$. 4. Calculate $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$ (because negative exponent flips the fraction). 5. Multiply the two results: $\frac{9}{4} \times \frac{9}{4} = \frac{81}{16}$. 6. The final answer is $\frac{81}{16}$. 3. **Simplify (a) $\frac{(g^2)^3}{g^{-9}} \times g^3$** 1. Use the power of a power rule: $(a^m)^n = a^{m \times n}$. 2. So, $(g^2)^3 = g^{2 \times 3} = g^6$. 3. The expression becomes $\frac{g^6}{g^{-9}} \times g^3$. 4. Division of powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$. 5. So, $\frac{g^6}{g^{-9}} = g^{6 - (-9)} = g^{6 + 9} = g^{15}$. 6. Now multiply by $g^3$: $g^{15} \times g^3 = g^{15 + 3} = g^{18}$. 7. The final simplified expression is $g^{18}$. 4. **Simplify (b) $\frac{6p^2q^3}{2(pq^{-1})^{-2}} \times 4pq$** 1. First simplify the denominator inside the parentheses: $(pq^{-1})^{-2}$. 2. Use the power of a product rule: $(ab)^n = a^n b^n$. 3. So, $(pq^{-1})^{-2} = p^{-2} (q^{-1})^{-2} = p^{-2} q^{2}$. 4. The denominator is $2 \times p^{-2} q^{2}$. 5. Rewrite the whole expression: $$\frac{6 p^2 q^3}{2 p^{-2} q^{2}} \times 4 p q$$ 6. Simplify the fraction coefficients: $\frac{6}{2} = 3$. 7. Simplify the powers of $p$: $p^{2} \div p^{-2} = p^{2 - (-2)} = p^{4}$. 8. Simplify the powers of $q$: $q^{3} \div q^{2} = q^{3 - 2} = q^{1} = q$. 9. So the fraction simplifies to $3 p^{4} q$. 10. Now multiply by $4 p q$: $3 p^{4} q \times 4 p q = (3 \times 4) p^{4 + 1} q^{1 + 1} = 12 p^{5} q^{2}$. 11. The final simplified expression is $12 p^{5} q^{2}$. **Summary:** 1(a) $5^3 \div (5^2 \times 5^{-3}) = 5^4 = 625$ 1(b) $\left(\frac{3}{2}\right)^2 \times \left(\frac{2}{3}\right)^{-2} = \frac{81}{16}$ 2(a) $(g^2)^3 \div g^{-9} \times g^3 = g^{18}$ 2(b) $6p^2q^3 \div 2(pq^{-1})^{-2} \times 4pq = 12 p^{5} q^{2}$