Simplify the Radical Expression square root of 8/25

Solution:

Step 1:

Express $\sqrt{\frac{8}{25}}$ as $\frac{\sqrt{8}}{\sqrt{25}}$.

Step 2:

Simplify the numerator.

Step 2.1:

Decompose $8$ into $2^2 \cdot 2$.

Step 2.1.1:

Extract $4$ from $8$ as $\frac{\sqrt{4 \cdot 2}}{\sqrt{25}}$.

Step 2.1.2:

Represent $4$ as $2^2$ to get $\frac{\sqrt{2^2 \cdot 2}}{\sqrt{25}}$.

Step 2.2:

Extract terms from under the radical, resulting in $\frac{2\sqrt{2}}{\sqrt{25}}$.

Step 2.3:

Recognize that the absolute value represents the non-negative magnitude of a number, hence $\frac{2\sqrt{2}}{\sqrt{25}}$ remains unchanged.

Step 3:

Simplify the denominator.

Step 3.1:

Express $25$ as $5^2$ to get $\frac{2\sqrt{2}}{\sqrt{5^2}}$.

Step 3.2:

Extract terms from under the radical, simplifying to $\frac{2\sqrt{2}}{5}$.

Step 3.3:

Since the absolute value of $5$ is $5$, the expression $\frac{2\sqrt{2}}{5}$ remains the same.

Step 4:

Present the final result in various formats.

Exact Form: $\frac{2\sqrt{2}}{5}$

Decimal Form: Approximately $0.56568542 \ldots$

Knowledge Notes:

• The square root of a quotient can be expressed as the quotient of the square roots of the numerator and the denominator: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

• To simplify a square root, factor the number under the radical into a product of squares and non-squares, then pull out the square terms.

• The absolute value of a number is its distance from zero on the number line, which is always non-negative.

• When extracting a square from under a square root, the result is the absolute value of the base of the square because square roots are defined to yield non-negative results.

• The square root of a perfect square, such as $4$ or $25$, is simply the base of the square ($2$ for $4$ and $5$ for $25$).

• The exact form of a simplified radical expression is preferred in mathematics because it maintains precision, whereas the decimal form is an approximation.