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How do you simplify $\sqrt{\dfrac{25}{64}}?$

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Last updated date: 24th Feb 2024
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IVSAT 2024
Answer
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Hint: Here, we have to simplify the given example $\sqrt{\dfrac{25}{64}}$ make the fraction in simplified form and take the square root of it to get the answer.

Complete step by step solution:
In the given example we have to find the square root of fraction. To simplify the given example which is $\sqrt{\dfrac{25}{64}}$ rewrite as $\dfrac{\sqrt{25}}{\sqrt{64}}$ by separating square root.
Now, firstly simplify the numerator which is $\sqrt{25}$ Rewrite $25$ as ${{5}^{2}}$ as we know that ${{5}^{2}}$ is equal to $25.$
Therefore, $\dfrac{\sqrt{25}}{\sqrt{64}}$ can be written as $\dfrac{\sqrt{52}}{\sqrt{64}}$
Now, pull the terms of the numerator out from the square root as we know that $\sqrt{{{5}^{2}}}$ is equal to $5.$Assuming the positive real numbers.
Therefore, we have $\dfrac{5}{\sqrt{64}}$
Now, simplifying denominator which is $\sqrt{64}.$
Rewrite $64$ as ${{8}^{2}}$ as we also know that ${{8}^{2}}$ is equal to $64.$
Therefore $\dfrac{5}{\sqrt{64}}$ can be written as $\dfrac{5}{\sqrt{{{8}^{2}}}}$
Now, pull the terms of denominator out from the square root as we know that $\sqrt{{{8}^{2}}}$ is equal to $8.$ Here, also therefore, we have $\dfrac{5}{8}$
The result can be shown in multiple forms rational from is $\dfrac{5}{8}$ and the decimal form is $0.625.$

Therefore, $\sqrt{\dfrac{25}{64}}=\dfrac{5}{8}=0.625$

Additional Information:
Square root of a number is a value which on multiplied by itself gives the original number here, $'\sqrt{{}}'$ is the radical symbol used to represent the root of the number. If we have to represent $'x'$ as a square root using this symbol can be written as $\sqrt{x}$. Where $x$ is the number. The number under the radical symbol is called a radicand. For example, the square root of a $8$ is also represented as radical of $8.$ Both represent the same value. The square root of a negative number represents a complex number. Suppose $\sqrt{-n}=i\sqrt{n}$, Where $'i'$ is the imaginary number.

Note: In this solution while taking the square root of $25$i.e. $\sqrt{25}$. It can give both positive and negative values that are $5$ and $5.$ But for solving this example. We have to assume only the positive values similarly for the square root of $64.$ i.e. $\sqrt{64}$ we assume positive value.
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