Higher Roots of Positive Numbers

In summary, the conversation discusses the concept of successive rooting operations on positive numbers and the potential existence of a synthesized formula to prove the property that the infinite root of any positive number is equal to one. It is also mentioned that the statement can be proven using various methods, such as using the exponential function or through an indirect proof. Ultimately, the conclusion is that the limit of successive rooting operations on any positive number is equal to one.
  • #1
Playing around with my calculator, I realized that if I do successive rooting operations on any positive non-zero number, I always get the number one.
Can I conclude that the infinite root of any positive number will always be zero?
If the statement is true, is there any synthesized formula to prove this property?

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  • #2
$$\lim_{n\to \infty} a^{1/n} = 1$$ for all positive a. You can show this e.g. using ##a^{1/n} = e^{\log(a)/n}## and continuity of the exponential function to bring the limit into the exponent, but you can also show it for successive square roots in more elementary ways by showing that e.g. the distance to 1 decreases at least by a factor 2 each time.
 
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  • #3
dom_quixote said:
Playing around with my calculator, I realized that if I do successive rooting operations on any positive non-zero number, I always get the number one.
Can I conclude that the infinite root of any positive number will always be zero?
No, you cannot conclude anything from trying out a couple of numbers. And there is no such thing like an infinite root. However, you can pose the hypothesis that
$$
\lim_{n \to \infty} \sqrt[n]{x} = 1 \text{ for }x>0
$$
dom_quixote said:
If the statement is true, is there any synthesized formula to prove this property?
You can define a sequence ##a_n:=\sqrt[n]{x}## and prove that ##|a_n-1|## get's arbitrary small for large ##n##.
 
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  • #4
mfb said:
$$\lim_{n\to \infty} a^{1/n} = 1$$ for all positive a. You can show this e.g. using ##a^{1/n} = e^{\log(a)/n}## and continuity of the exponential function to bring the limit into the exponent, but you can also show it for successive square roots in more elementary ways by showing that e.g. the distance to 1 decreases at least by a factor 2 each time.
You can also do an indirect proof. For any ##a > 1##, the sequence ##a^{1/n}## is monotone decreasing and bounded below by ##1##. Therefore, the limit exists.

However, for any number ##l > 1##, the sequence ##l^n## in unbounded; and so ##l \ne lim_{n \to \infty} a^{1/n}##. That leaves the only possibility that ##lim_{n \to \infty} a^{1/n} = 1##.
 
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What are higher roots of positive numbers?

Higher roots of positive numbers refer to the roots of a number that are greater than the square root. For example, the cube root (3rd root) of 8 is 2, and the fourth root of 16 is 2.

How do you calculate higher roots?

To calculate higher roots, you can use the power or exponent notation. For example, the cube root of 8 can be written as 8^(1/3) or 8^(0.333). You can also use a calculator or an online calculator to find the higher root of a number.

What is the difference between higher roots and square roots?

The main difference between higher roots and square roots is the degree of the root. Square roots are the 2nd root, while higher roots refer to any root that is greater than 2. Additionally, square roots are commonly used in finding the length of sides in geometric shapes, while higher roots are used in various mathematical calculations.

What are some real-life applications of higher roots?

Higher roots have various applications in fields such as engineering, physics, and finance. For example, the cube root is used in calculating the volume of a cube, the fourth root is used in calculating the area of a circle, and the fifth root is used in calculating the interest rate for compound interest.

What are some common misconceptions about higher roots?

One common misconception is that higher roots are only applicable to positive numbers. However, higher roots can also be calculated for negative numbers, imaginary numbers, and complex numbers. Another misconception is that higher roots always result in a whole number, when in fact, they can result in fractions or irrational numbers.

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