How do you find the square root of 130 in radical form?
The Correct Answer and Explanation is:
To find the square root of ( 130 ) in radical form, we write it as ( \sqrt{130} ). Since ( 130 ) is not a perfect square, its square root cannot be simplified to an integer. However, we can check if it can be simplified by factoring ( 130 ) into its prime factors.
Step-by-Step Process:
- Factorize ( 130 ):
- The prime factorization of ( 130 ) is ( 130 = 2 \cdot 5 \cdot 13 ).
- None of the factors ( 2, 5, ) or ( 13 ) are perfect squares.
- Simplify the Radical:
- In radical form, ( \sqrt{130} ) does not simplify further because no pair of identical prime factors exists to “pull out” of the radical.
- Verify Approximation:
- ( \sqrt{130} ) is between ( \sqrt{121} = 11 ) and ( \sqrt{144} = 12 ), so its approximate value is slightly above 11.
- Using a calculator, ( \sqrt{130} \approx 11.4018 ).
- Conclusion:
- The exact square root of ( 130 ) in radical form remains ( \sqrt{130} ).
Detailed Explanation:
Radical simplification involves expressing a square root in its simplest form by identifying and factoring out perfect squares. For example, ( \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} ). This process is not possible with ( \sqrt{130} ) because its prime factors ( 2, 5, ) and ( 13 ) do not form any perfect square products.
In general, a square root that cannot be simplified further is called an “irrational number.” It remains in radical form, like ( \sqrt{130} ), or is approximated using decimal values. While irrational numbers cannot be written exactly as decimals, they are crucial in mathematics, especially in geometry and algebra.
Thus, the square root of ( 130 ) in radical form is ( \sqrt{130} ), and it does not simplify further.