What is the square root of [tex]$\sqrt{90}$[/tex]?
Answer (2)
The square root of the square root of 90, represented as 90 , is simplified to 9 0 4 1 , which is approximately 3.162. This is done by rewriting the expression using exponents and calculating the fourth root. The precise value is essential for applications in mathematics and various scientific fields.
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Rewrite the expression as 9 0 4 1 .
Calculate the value of 9 0 4 1 .
The result is approximately 3.080.
The square root of the square root of 90 is 3.080 .
Explanation
Understanding the problem We are asked to find the square root of the square root of 90. This can be written mathematically as 90 .
Rewriting with exponents To simplify this expression, we can rewrite it using exponents. Recall that x = x 2 1 . Therefore, 90 = ( 9 0 2 1 ) 2 1 .
Simplifying the exponent Using the property of exponents, ( a m ) n = a m × n , we have ( 9 0 2 1 ) 2 1 = 9 0 2 1 × 2 1 = 9 0 4 1 .
Calculating the fourth root Now, we can calculate the value of 9 0 4 1 . This is the same as finding the fourth root of 90. The result of this operation is approximately 3.080.
Final Answer Therefore, the square root of the square root of 90 is approximately 3.080.
Examples
Understanding nested roots is useful in various fields, such as signal processing and data compression, where signals or data may undergo multiple transformations. For instance, in image processing, a signal might be transformed multiple times to extract features or reduce noise. Calculating nested roots helps in analyzing the overall effect of these transformations and understanding the underlying data characteristics. This concept is also applicable in financial modeling, where returns on investments might be calculated over multiple time periods, requiring the analysis of nested roots to determine overall growth rates.
