Combine and simplify the following radical expression:
[tex]$\sqrt{\frac{25}{128}}$[/tex]
Answer (2)
To simplify 128 25 , we separate it into 128 25 , which simplifies to 8 2 5 . After rationalizing the denominator, we find the final result to be 16 5 2 .
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Rewrite the expression as a fraction of square roots: 128 25 = 128 25 .
Simplify the numerator: 128 25 = 128 5 .
Simplify the denominator: 128 = 8 2 .
Rationalize the denominator: 8 2 5 = 16 5 2 .
The simplified expression is 16 5 2 .
Explanation
Understanding the Problem We are given the radical expression 128 25 . Our goal is to simplify this expression.
Separating the Radical First, we can rewrite the expression as a fraction of square roots: 128 25 = 128 25 .
Simplifying the Numerator The square root of 25 is 5, so we have: 128 25 = 128 5 .
Simplifying the Denominator Now, we need to simplify the denominator. We can factor 128 as 128 = 2 7 = 2 6 ⋅ 2 = ( 2 3 ) 2 ⋅ 2 = 64 ⋅ 2 . Therefore, we can rewrite the denominator as: 128 = 64 ⋅ 2 = 64 ⋅ 2 = 8 2 .
Substituting the Simplified Denominator So, our expression becomes: 128 5 = 8 2 5 .
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 2 : 8 2 5 = 8 2 ⋅ 2 5 ⋅ 2 = 8 ⋅ 2 5 2 = 16 5 2 .
Final Answer Therefore, the simplified expression is 16 5 2 .
Examples
Radical expressions are useful in various fields, such as engineering and physics, when dealing with lengths, areas, and volumes. For instance, when calculating the diagonal of a square with side length s , the diagonal is s 2 . Simplifying radical expressions allows for easier calculations and a better understanding of the relationships between different quantities. Consider a scenario where you need to determine the length of a support beam in a bridge design, and the calculation involves simplifying a radical expression to ensure structural integrity.
