Answer :
Let's simplify each part step-by-step:
### Part (i): Simplify \(3\left(a^4 b^3\right)^{10} \times 5\left(a^2 b^2\right)^3\)
1. Apply the exponentiation rule \((a^m)^n = a^{mn}\):
[tex]\[ 3\left(a^4 b^3\right)^{10} = 3 \cdot \left(a^{4 \cdot 10} \cdot b^{3 \cdot 10}\right) = 3 \cdot a^{40} \cdot b^{30} \][/tex]
[tex]\[ 5\left(a^2 b^2\right)^3 = 5 \cdot \left(a^{2 \cdot 3} \cdot b^{2 \cdot 3}\right) = 5 \cdot a^6 \cdot b^6 \][/tex]
2. Multiply the resulting expressions:
[tex]\[ 3 \cdot a^{40} \cdot b^{30} \times 5 \cdot a^6 \cdot b^6 \][/tex]
3. Combine the constants:
[tex]\[ 3 \cdot 5 = 15 \][/tex]
4. Use the product rule for exponents \((a^m \cdot a^n = a^{m+n})\) to combine like terms:
[tex]\[ a^{40} \times a^6 = a^{40 + 6} = a^{46} \][/tex]
[tex]\[ b^{30} \times b^6 = b^{30 + 6} = b^{36} \][/tex]
5. Final simplified expression:
[tex]\[ 15 \cdot a^{46} \cdot b^{36} \][/tex]
So, the simplified form of part (i) is \(15a^{46}b^{36}\).
### Part (ii): Simplify \(\left(2 x^{-2} y^3\right)^3\)
1. Apply the exponentiation rule \((a^m)^n = a^{mn}\):
[tex]\[ \left(2 x^{-2} y^3\right)^3 = 2^3 \cdot \left(x^{-2}\right)^3 \cdot \left(y^3\right)^3 \][/tex]
2. Simplify each factor separately:
[tex]\[ 2^3 = 8 \][/tex]
[tex]\[ \left(x^{-2}\right)^3 = x^{-2 \cdot 3} = x^{-6} \][/tex]
[tex]\[ \left(y^3\right)^3 = y^{3 \cdot 3} = y^9 \][/tex]
3. Combine all simplified factors:
[tex]\[ 8 \cdot x^{-6} \cdot y^9 \][/tex]
4. Express the final result:
[tex]\[ 8 \cdot \frac{y^9}{x^6} \][/tex]
So, the simplified form of part (ii) is \(8 \cdot \frac{y^9}{x^6}\) or \(8y^9 x^{-6}\).
### Final Results
(i) \( 3\left(a^4 b^3\right)^{10} \times 5\left(a^2 b^2\right)^3 \) simplifies to \( 15a^{46}b^{36} \)
(ii) [tex]\( \left(2 x^{-2} y^3\right)^3 \)[/tex] simplifies to [tex]\( 8 \cdot \frac{y^9}{x^6} \)[/tex] or [tex]\( 8y^9 x^{-6} \)[/tex]
### Part (i): Simplify \(3\left(a^4 b^3\right)^{10} \times 5\left(a^2 b^2\right)^3\)
1. Apply the exponentiation rule \((a^m)^n = a^{mn}\):
[tex]\[ 3\left(a^4 b^3\right)^{10} = 3 \cdot \left(a^{4 \cdot 10} \cdot b^{3 \cdot 10}\right) = 3 \cdot a^{40} \cdot b^{30} \][/tex]
[tex]\[ 5\left(a^2 b^2\right)^3 = 5 \cdot \left(a^{2 \cdot 3} \cdot b^{2 \cdot 3}\right) = 5 \cdot a^6 \cdot b^6 \][/tex]
2. Multiply the resulting expressions:
[tex]\[ 3 \cdot a^{40} \cdot b^{30} \times 5 \cdot a^6 \cdot b^6 \][/tex]
3. Combine the constants:
[tex]\[ 3 \cdot 5 = 15 \][/tex]
4. Use the product rule for exponents \((a^m \cdot a^n = a^{m+n})\) to combine like terms:
[tex]\[ a^{40} \times a^6 = a^{40 + 6} = a^{46} \][/tex]
[tex]\[ b^{30} \times b^6 = b^{30 + 6} = b^{36} \][/tex]
5. Final simplified expression:
[tex]\[ 15 \cdot a^{46} \cdot b^{36} \][/tex]
So, the simplified form of part (i) is \(15a^{46}b^{36}\).
### Part (ii): Simplify \(\left(2 x^{-2} y^3\right)^3\)
1. Apply the exponentiation rule \((a^m)^n = a^{mn}\):
[tex]\[ \left(2 x^{-2} y^3\right)^3 = 2^3 \cdot \left(x^{-2}\right)^3 \cdot \left(y^3\right)^3 \][/tex]
2. Simplify each factor separately:
[tex]\[ 2^3 = 8 \][/tex]
[tex]\[ \left(x^{-2}\right)^3 = x^{-2 \cdot 3} = x^{-6} \][/tex]
[tex]\[ \left(y^3\right)^3 = y^{3 \cdot 3} = y^9 \][/tex]
3. Combine all simplified factors:
[tex]\[ 8 \cdot x^{-6} \cdot y^9 \][/tex]
4. Express the final result:
[tex]\[ 8 \cdot \frac{y^9}{x^6} \][/tex]
So, the simplified form of part (ii) is \(8 \cdot \frac{y^9}{x^6}\) or \(8y^9 x^{-6}\).
### Final Results
(i) \( 3\left(a^4 b^3\right)^{10} \times 5\left(a^2 b^2\right)^3 \) simplifies to \( 15a^{46}b^{36} \)
(ii) [tex]\( \left(2 x^{-2} y^3\right)^3 \)[/tex] simplifies to [tex]\( 8 \cdot \frac{y^9}{x^6} \)[/tex] or [tex]\( 8y^9 x^{-6} \)[/tex]
