Write as a radical power
1. 3√13^2
2. √(2/3)^5
√ 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.
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
1. To write 3√13^2 as a radical power, we need to simplify the expression inside the radical first.
First, calculate 13^2, which equals 169.
Now, we can rewrite the expression as 3√169.
Since the index of the radical is 3, we need to find the cube root of 169.
The cube root of 169 is 5 because 5 * 5 * 5 = 125 and 6 * 6 * 6 = 216. So the closest whole number cube to 169 is 125, which is 5^3.
Therefore, 3√13^2 can be rewritten as 5.
So, 3√13^2 = 5.
2. To write √(2/3)^5 as a radical power, we simplify the expression inside the square root first.
First, calculate (2/3)^5, which is equal to 32/243.
Now, we can rewrite the expression as √(32/243).
To simplify further, we can break down 32 and 243 into their prime factorization:
32 = 2^5
243 = 3^5
So, √(32/243) becomes √((2^5)/(3^5)).
Next, we can simplify the fraction by cancelling out common factors within the square root. In this case, both the numerator and denominator have a factor of 2^5.
Thus, we can simplify further to get √(1/(3^5/2^5)).
Simplifying the fraction 3^5/2^5, we get (3/2)^5.
Therefore, √(2/3)^5 can be rewritten as √(1/(3/2)^5).
Now, since the square root and the fifth power are inverse operations, we can rewrite it as (3/2)^(5/2).
Thus, √(2/3)^5 can be expressed as (3/2)^(5/2).