Using the exponent properties, which of the following expressions is equivalent to (2–√)^3⋅2−^1/2?(1 point)
Responses
2
2
2–√
Start Root 2 End Root
212
2 superscript Start Fraction 1 over 2 End Fraction baseline
22
To simplify the expression (2–√)^3⋅2−^1/2 using exponent properties, we can apply the rule that states (a^m)^n = a^(m*n).
First, let's simplify (2–√)^3:
(2–√)^3 = (2–√)(2–√)(2–√)
Using the distributive property, we can expand this expression:
= (2–√) * (2–√ * 2–√)
= (2–√)*(2–√ * 2)
= (2–√)*(2–2√)
= (2–√)*(2 – 2√ + 2√ – 2) (using the distributive property again)
= (2–√)*(0)
= 0
Now let's simplify 2^(-1/2):
2^(-1/2) = 1/(2^(1/2)) = 1/√2
Therefore, the expression (2–√)^3⋅2−^1/2 simplifies to:
0 * 1/√2 = 0
So the correct response is 0.
To simplify the expression (2–√)^3⋅2−^1/2 using the exponent properties, we can apply the following rules:
1. (a^b)^c = a^(b*c) - This property allows us to raise a power to another power.
2. a^(-b) = 1/a^b - This property allows us to convert a negative exponent into a fraction.
Using these properties, let's simplify the given expression step by step:
Step 1: Simplify (2–√)^3 as (2^3)⋅(√2)^3.
(2–√)^3 = (2^3)⋅(√2)^3
= 8⋅(√2)^3
Step 2: Simplify (√2)^3 as (√2⋅√2⋅√2).
(√2)^3 = (√2⋅√2⋅√2)
= 2√2√2
= 2√4
Step 3: Simplify 2√4 as 2⋅2.
2√4 = 2⋅2
= 4
Step 4: Simplify 2−^1/2 as 1/(2^1/2) using property 2.
2−^1/2 = 1/(2^1/2)
= 1/√2
Step 5: Substitute the simplified values back into the original expression.
(2–√)^3⋅2−^1/2 = 8⋅(√2)^3⋅2−^1/2
= 8⋅4⋅1/√2
= 32/√2
To further simplify the expression, we can rationalize the denominator (√2) by multiplying the numerator and denominator by √2:
32/√2 = (32√2)/(√2⋅√2)
= (32√2)/2
= 16√2
Therefore, the equivalent expression is 16√2.