Question

1. What is the value of sqrt(63) * cube root of 56 * 7^(1/6)

I know the answer is 42, but I don't know how to solve it!

Answer 1

`√(63)* \sqrt[3]{56}*7^{(1)/(6)}=42`

Explanation:

Given : Expression
`√(63)* \sqrt[3]{56}*7^{(1)/(6)}`

To find : What is the value of the expression ?

Solution :

Step 1 - Write the expression as

`(63)^{(1)/(2)}*(56)^{(1)/(3)}*7^{(1)/(6)}`

Step 2 - Factor 63 and 56 as multiple of 7,

`(7* 9)^{(1)/(2)}*(7* 8)^{(1)/(3)}*7^{(1)/(6)}`

Step 3 - Split the powers
`(a* b)^c=a^c* b^c`

`7^{(1)/(2)}* 9^{(1)/(2)}*7^{(1)/(3)}* 8^{(1)/(3)}*7^{(1)/(6)}`

Step 4 - Add the same base power,
`a^b* a^c=a^(b+c)`

`=7^{(1)/(2)+(1)/(3)+(1)/(6)}* (3^2)^{(1)/(2)}* (2^3)^{(1)/(3)}`

`=7^{(3+2+1)/(6)}* (3)^{(2)/(2)}* (2)^{(3)/(3)}`

`=7^{(6)/(6)}* (3)^(1)* (2)^(1)`

`=7*3*2`

`=42`

Therefore,
`√(63)* \sqrt[3]{56}*7^{(1)/(6)}=42`