[tex]$\sqrt{98}$[/tex]
Answer (2)
To simplify 98 , we can break it down to its prime factors, which gives us 7 2 . This involves separating the square root into the square root of 2 and the square root of 7 2 , leading to the final simplified expression. The process illustrates the use of prime factorization and square root properties effectively.
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Find the prime factorization of 98: 98 = 2 × 7 2 .
Rewrite the square root: 98 = 2 × 7 2 .
Separate the factors: 2 × 7 2 = 2 × 7 2 .
Simplify: 98 = 7 2 .
Explanation
Understanding the Problem We are given the expression 98 and our goal is to simplify it.
Prime Factorization First, we need to find the prime factorization of 98. We can write 98 as 2 × 49 . Since 49 is 7 2 , the prime factorization of 98 is 2 × 7 2 .
Rewriting the Square Root Now we can rewrite the square root of 98 as the square root of its prime factors: 98 = 2 × 7 2 .
Separating Factors Using the property of square roots, we can separate the factors: 2 × 7 2 = 2 × 7 2 .
Simplifying Since 7 2 = 7 , we have 98 = 2 × 7 = 7 2 .
Final Answer Therefore, the simplified form of 98 is 7 2 .
Examples
Square roots are used in many areas of math and science. For example, when calculating the distance between two points in a coordinate plane, you use the distance formula, which involves square roots. If you have two points (1, 2) and (4, 6), the distance between them is ( 4 − 1 ) 2 + ( 6 − 2 ) 2 = 3 2 + 4 2 = 9 + 16 = 25 = 5 . Simplifying square roots helps in finding exact distances.
