Using the exponent properties, which of the following expressions is equivalent to (√2)^3⋅2^−1/2?(1 point)
Responses
2^2
√2
261/2
2
The correct answer is 2.
To solve this problem using exponent properties, we can rewrite (√2)^3 as (√2)^(2/2) and rewrite 2^(-1/2) as 2^(1/2)^(-1).
Then, (√2)^3⋅2^(-1/2) becomes (√2)^(2/2) * 2^(1/2)^(-1).
By applying the exponent property (a^m * a^n = a^(m+n)), this expression simplifies to (√2)^(2/2 + 1/2 * -1).
Since 2/2 + 1/2 * -1 simplifies to 1, we have (√2)^1.
Finally, any number raised to the power of 1 is equal to the number itself, so (√2)^1 is equal to √2.
Therefore, the equivalent expression is √2, option 2.
To simplify the expression (√2)^3⋅2^(-1/2), we can use the exponent properties.
First, let's simplify (√2)^3. To raise a square root to a power, we can square both the number inside the square root and the exponent. So, (√2)^3 = (2^(1/2))^3 = 2^(1/2 * 3) = 2^(3/2).
Now, let's simplify 2^(-1/2). The exponent -1/2 means we take the reciprocal of the square root of 2 raised to the power of 1/2. So, 2^(-1/2) = 1/(2^(1/2)) = 1/√2.
Putting it all together, we have (√2)^3⋅2^(-1/2) = 2^(3/2) * (1/√2) = 2^(3/2) / √2.
To simplify further, we can combine the fractions under a common denominator. The square root of 2 can be written as √2 = 2^(1/2), so the expression becomes (2^(3/2)) / (2^(1/2)).
Using the property of division with the same base, we subtract the exponents: 2^(3/2 - 1/2) = 2^(2/2) = 2^1 = 2.
Therefore, the expression (√2)^3⋅2^(-1/2) is equivalent to 2.