List three different ways to simplify (sqrt(12))^2. Provide an explanation for how each expression is equal to (sqrt(12))^2.
To simplify the expression (sqrt(12))^2, which represents squaring the square root of 12, we can use three different approaches:
1. Algebraic simplification:
Start with (sqrt(12))^2.
Applying the power rule for exponents, the square is distributed to both the base and the exponent.
(sqrt(12))^2 simplifies to 12, since the square of the square root of 12 is equal to 12.
2. Radical simplification:
Start with (sqrt(12))^2.
The square root and square operations are inverse operations of each other, which means they cancel each other out.
(sqrt(12))^2 simplifies to 12, because the square and square root cancel each other out, leaving us with the original number.
3. Arithmetic simplification:
Start with (sqrt(12))^2.
Evaluate the square root of 12, which is approximately equal to 3.464.
Squaring 3.464 gives us approximately 12, since the square of the square root of 12 is equal to 12 (as shown in the first two approaches).
In summary, all three expressions (algebraic simplification, radical simplification, and arithmetic simplification) result in the value 12 for the expression (sqrt(12))^2.