Question

What is the following product ^3sqrt16x^7*^3sqrt12x^9

Answer 1

The expression
`\sqrt[3]{16x^(7) } * \sqrt[3]{12x^(9) }` =
`4x^(5)} (\sqrt[3]{3x})`

Explanation:

Given

`\sqrt[3]{16x^(7) } * \sqrt[3]{12x^(9) }`

Required

Products of both

To do this, we have to apply the laws of indices,

Follow the highlighted steps

Step 1: Multiply both parameters directly

Since they both have the same roots, they can be multiplied directly according to the law of indices

`\sqrt[3]{16x^(7) } * \sqrt[3]{12x^(9) }` becomes

`\sqrt[3]{16x^(7) * 12x^(9) }`

Step 2: Apply the 1st law of indices

First law of indices states that

`x^(a) * x^(b) = x^(a + b)`

So,
`\sqrt[3]{16x^(7) * 12x^(9) }` becomes

`\sqrt[3]{16x^(7) * 12x^(9) }` =
`\sqrt[3]{16 * 12 * x^(7) * x^(9) }`

`\sqrt[3]{16x^(7) * 12x^(9) }` =
`\sqrt[3]{16 * 12 * x^(7+9) }`

`\sqrt[3]{16x^(7) * 12x^(9) }` =
`\sqrt[3]{16 * 12 * x^(16) }`

`\sqrt[3]{16x^(7) * 12x^(9) }` =
`\sqrt[3]{192 * x^(16) }`

Step 3: Rewrite the expression

`\sqrt[3]{192 * x^(16) }` =
`({192 * x^(16) })^{(1)/(3) }`

Step 4: Expand the Expression in bracket

`({192 * x^(16) })^{(1)/(3) }` =
`({64 * 3* x^(15) * x^(1) })^{(1)/(3) }`

Break down into bits

`({192 * x^(16) })^{(1)/(3) }` =
`64^(1)/(3) * 3^(1)/(3) * (x^(15))^(1)/(3) * (x^(1))(1)/(3)`

`({192 * x^(16) })^{(1)/(3) }` =
`(4^(3)) ^(1)/(3) * 3^(1)/(3) * (x^(15))^(1)/(3) * (x^(1))(1)/(3)`

`({192 * x^(16) })^{(1)/(3) }` =
`(4^{3*(1)/(3)}) * 3^(1)/(3) * (x^(15)*^(1)/(3)) * (x^{(1)/(3)})`

`({192 * x^(16) })^{(1)/(3) }` =
`4 * 3^(1)/(3) * (x^(5)}) * (x^{(1)/(3)})`

`({192 * x^(16) })^{(1)/(3) }` =
`4 (x^(5)})* 3^(1)/(3) * (x^{(1)/(3)})`

`({192 * x^(16) })^{(1)/(3) }` =
`4x^(5)} * (3^(1)/(3) * x^{(1)/(3)})`

`({192 * x^(16) })^{(1)/(3) }` =
`4x^(5)} * (3x)^(1)/(3)`

`({192 * x^(16) })^{(1)/(3) }` =
`4x^(5)} * \sqrt[3]{3x}`

`({192 * x^(16) })^{(1)/(3) }` =
`4x^(5)} (\sqrt[3]{3x})`

Hence, the expression
`\sqrt[3]{16x^(7) } * \sqrt[3]{12x^(9) }` =
`4x^(5)} (\sqrt[3]{3x})`

Answer 2

4x^5\sqrt[3]{3x}

Explanation:

Given

`\sqrt[3]{16x^7}\left(\sqrt[3]{12x^9}\right)\\\sqrt[3]{4^(2) x^7}\sqrt[3]{4(3)x^9}\\  4^(2)/(3) . x^(7)/(3) . 4^(1)/(3) . 3^(1)/(3) . x^(9)/(3)\\  4^(2+1)/(3) .  3^(1)/(3). x^(9+7)/(3) \\4^(3)/(3) . 3^(1)/(3). x^(16)/(3)\\ 4\sqrt[3]{3x^16}`

`4\sqrt[3]{3} . x^(16)/(3) \\4\sqrt[3]{3} . x^(15)/(3) .x^(1)/(3) \\4\sqrt[3]{3} . x^5 .x^(1)/(3) \\4x^5\sqrt[3]{3x}` !