Question

Which expression is equivalent to 3√64ab²c³?

2abc²[√4a²b³c]
4a²b²c³ (3√5)
8a³b³c¹ (3√/bc)
8a²b²c³(3√/b)

Answer 1

• ³√64ab²c³
• √4³ab²c³
• 4c√a^{2/3}b^{2/3}

The expression is seemed to have none of the above solutions

Answer 2

`4a^2b^2c^3\left(\sqrt[3]{b}\right)`

Explanation:

**Please note that the expression quoted in the question is likely incorrect (see attachment)**

Assuming the expression is:

`\sqrt[3]{64a^6b^7c^9}`

`\textsf{Apply radical rule} \quad √(ab)=√(a){ \cdot √(b)`

`\implies \sqrt[3]{64} \cdot \sqrt[3]{a^6}\cdot \sqrt[3]{b^7}\cdot \sqrt[3]{c^9}`

Rewrite 64 as 4³:

`\implies \sqrt[3]{4^3} \cdot \sqrt[3]{a^6}\cdot \sqrt[3]{b^7}\cdot \sqrt[3]{c^9}`

`\textsf{Apply exponent rule} \quad a^(b+c)=a^b \cdot a^c \quad \sf to\:\:b^7`

`\implies b^7=b^(6+1)=b^6b^1=b^6b`

Therefore:

`\implies \sqrt[3]{4^3} \cdot \sqrt[3]{a^6}\cdot \sqrt[3]{b^(6)b}\cdot \sqrt[3]{c^9}`

`\implies \sqrt[3]{4^3} \cdot \sqrt[3]{a^6}\cdot \sqrt[3]{b^(6)}\cdot \sqrt[3]{b}\cdot \sqrt[3]{c^9}`

`\textsf{Apply exponent rule} \quad \sqrt[n]{a^m}=a^{(m)/(n)}`

`\implies 4^{(3)/(3)} \cdot a^{(6)/(3)} \cdot b^{(6)/(3)} \cdot \sqrt[3]{b} \cdot c^{(9)/(3)}`

Simplify:

`\implies 4^1 \cdot a^2 \cdot b^2 \cdot \sqrt[3]{b} \cdot c^3`

`\implies 4a^2b^2c^3\left(\sqrt[3]{b}\right)`