Simplify square root of (b^4)/25

Solution:

Step 1:

Express $b^{4}$ as $(b^{2})^{2}$. Now the expression is $\sqrt{\frac{(b^{2})^{2}}{25}}$.

Step 2:

Represent $25$ as $5^{2}$. The expression becomes $\sqrt{\frac{(b^{2})^{2}}{5^{2}}}$.

Step 3:

Reformulate $\frac{(b^{2})^{2}}{5^{2}}$ as $\left(\frac{b^{2}}{5}\right)^{2}$. Our expression is now $\sqrt{\left(\frac{b^{2}}{5}\right)^{2}}$.

Step 4:

Extract terms from the square root, assuming all numbers are positive real numbers. The simplified form is $\frac{b^{2}}{5}$.

Knowledge Notes:

To simplify a square root of a fraction, we can use the property of square roots that states $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. Additionally, the square root of a square number is the base of the square, provided that we are working with positive real numbers. This is because the square root and the square are inverse operations.

1. Square of a Square: The square of a square, such as $(b^{2})^{2}$, simplifies to $b^{4}$ because the exponents are multiplied in this case.

2. Square Root of a Square: The square root of a square, such as $\sqrt{(b^{2})^{2}}$, simplifies to $b^{2}$, assuming $b$ is a positive real number.

3. Square Root of a Fraction: The square root of a fraction can be simplified by taking the square root of the numerator and the denominator separately, as in $\sqrt{\frac{a^{2}}{b^{2}}} = \frac{\sqrt{a^{2}}}{\sqrt{b^{2}}} = \frac{a}{b}$.

4. Simplifying Square Roots: When simplifying square roots, it is often helpful to express the terms under the radical in their square forms, which allows for easier simplification.

5. Assumption of Positive Real Numbers: When simplifying expressions involving square roots, it is common to assume that the variables represent positive real numbers to avoid dealing with complex numbers or the absolute value function.