√ means the 1/2 power.
As with any exponents, powers of powers multiply exponents.
√(3^5) = (3^5)^(1/2) = 3^(5/2)
(√3)^5 = (3^(1/2))^5 = 3^(5/2)
Now try your problems.
Write as a radical power
1. 3√13^2
2. √(2/3)^5
2 answers
1.
√ 13 ^ 2 = 13
3 √ 13 ^ 2 = 3 * 13 = 39
2.
( 2 / 3 ) ^ 5 = ( 2 / 3 ) ^ 4 * ( 2 / 3 ) =
[ ( 2 / 3 ) ^ 2 ] ^ 2 * ( 2 / 3 )
√ ( 2 / 3 ) ^ 5 = √ [ ( 2 / 3 ) ^ 2 ) ^ 2 ] * √ ( 2 / 3 ) =
( 2 / 3 ) ^ 2 * √ ( 2 / 3 ) =
( 4 / 9 ) * √ ( 2 / 3 ) =
4 * √ ( 2 / 3 ) / 9 =
4 * √ 2 / 9 * √ 3
√ 13 ^ 2 = 13
3 √ 13 ^ 2 = 3 * 13 = 39
2.
( 2 / 3 ) ^ 5 = ( 2 / 3 ) ^ 4 * ( 2 / 3 ) =
[ ( 2 / 3 ) ^ 2 ] ^ 2 * ( 2 / 3 )
√ ( 2 / 3 ) ^ 5 = √ [ ( 2 / 3 ) ^ 2 ) ^ 2 ] * √ ( 2 / 3 ) =
( 2 / 3 ) ^ 2 * √ ( 2 / 3 ) =
( 4 / 9 ) * √ ( 2 / 3 ) =
4 * √ ( 2 / 3 ) / 9 =
4 * √ 2 / 9 * √ 3