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anemone
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Prove that for any natural number $a$, there is an integer $k$ such that $(\sqrt{1982}+1)^a=\sqrt{k}+\sqrt{k+1981^a}$
Surds are numbers that cannot be expressed as exact decimals or fractions. They usually involve irrational numbers, such as the square root of 2 or pi.
Powers and radicals are inverse operations. A power raises a number to a given exponent, while a radical takes the root of a number. For example, 2^3 (2 raised to the power of 3) is equal to 8, while √8 (square root of 8) is also equal to 8.
To simplify a surd, you need to factor the number inside the square root sign and look for perfect squares. Then, you can take the square root of those perfect squares and leave the remaining numbers outside the square root sign. For example, the surd √36 can be simplified to 6, as 6 is a perfect square.
A rational number is one that can be expressed as a ratio of two integers, while an irrational number cannot be expressed as a ratio of two integers. Irrational numbers, such as pi or the square root of 2, have non-terminating and non-repeating decimals.
To perform operations with surds, you need to follow the same rules as with other numbers. For addition and subtraction, you can only combine like terms (those with the same surd). For multiplication, you can multiply the numbers outside the square root sign and the numbers inside the square root sign separately. For division, you can rationalize the denominator by multiplying both the numerator and denominator by the surd in the denominator.