Question
Apply all the properties of integer exponents to generate equivalent expressions to with the power of two over two minus power 4 we only positive exponents and solve the expression provide the solution as a whole number
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GPT 3.5
To generate equivalent expressions, we can apply the properties of integer exponents.
Given expression: (2^2)/2^4
Using the properties of exponents:
1. Power of a Power: (a^m)^n = a^(m * n)
(2^2) / (2^4) can be simplified as 2^(2 * 1/4).
2. Power of a Quotient: (a/b)^n = (a^n)/(b^n)
2^(2 * 1/4) can be written as (2^2)^(1/4) / (2^4)^(1/4).
3. Simplifying the exponents separately:
(2^2)^(1/4) = 2^(2 * 1/4) = 2^(1/2) = √2
(2^4)^(1/4) = 2^(4 * 1/4) = 2^(1) = 2
Therefore, the expression becomes √2 / 2.
To solve the expression √2 / 2 as a whole number, we can multiply the numerator and denominator by √2:
(√2 * √2) / (2 * √2) = 2 / (2 * √2) = 1 / √2
To rationalize the denominator, we can multiply the numerator and denominator by √2:
(1 * √2) / (√2 * √2) = √2 / 2
The solution, expressed as a whole number, is √2 / 2.
Given expression: (2^2)/2^4
Using the properties of exponents:
1. Power of a Power: (a^m)^n = a^(m * n)
(2^2) / (2^4) can be simplified as 2^(2 * 1/4).
2. Power of a Quotient: (a/b)^n = (a^n)/(b^n)
2^(2 * 1/4) can be written as (2^2)^(1/4) / (2^4)^(1/4).
3. Simplifying the exponents separately:
(2^2)^(1/4) = 2^(2 * 1/4) = 2^(1/2) = √2
(2^4)^(1/4) = 2^(4 * 1/4) = 2^(1) = 2
Therefore, the expression becomes √2 / 2.
To solve the expression √2 / 2 as a whole number, we can multiply the numerator and denominator by √2:
(√2 * √2) / (2 * √2) = 2 / (2 * √2) = 1 / √2
To rationalize the denominator, we can multiply the numerator and denominator by √2:
(1 * √2) / (√2 * √2) = √2 / 2
The solution, expressed as a whole number, is √2 / 2.