Simplify square root of 1 24/25

Solution:

Step:1 Transform the mixed number $1\frac{24}{25}$ into an improper fraction.

Step:1.1 To convert a mixed number, sum the whole number part and the fraction part: $\sqrt{1+\frac{24}{25}}$

Step:1.2 Perform the addition of $1$ and $\frac{24}{25}$.

Step:1.2.1 Express $1$ as a fraction with the same denominator: $\sqrt{\frac{25}{25}+\frac{24}{25}}$

Step:1.2.2 Sum the numerators while keeping the denominator the same: $\sqrt{\frac{25+24}{25}}$

Step:1.2.3 Calculate the sum of $25$ and $24$: $\sqrt{\frac{49}{25}}$

Step:2 Express $\sqrt{\frac{49}{25}}$ as the quotient of square roots: $\frac{\sqrt{49}}{\sqrt{25}}$.

Step:3 Simplify the square root in the numerator.

Step:3.1 Represent $49$ as a square of $7$: $\frac{\sqrt{7^2}}{\sqrt{25}}$

Step:3.2 Extract the square root of the perfect square: $\frac{7}{\sqrt{25}}$

Step:4 Simplify the square root in the denominator.

Step:4.1 Express $25$ as a square of $5$: $\frac{7}{\sqrt{5^2}}$

Step:4.2 Extract the square root of the perfect square: $\frac{7}{5}$

Step:5 Present the result in various formats.

Exact Form: $\frac{7}{5}$ Decimal Form: $1.4$ Mixed Number Form: $1\frac{2}{5}$

Knowledge Notes:

The process of simplifying the square root of a mixed number involves several steps:

1. Conversion to an Improper Fraction: A mixed number is composed of a whole number and a fraction. To work with it easily, it's converted to an improper fraction, which is a fraction where the numerator is greater than or equal to the denominator.

2. Simplifying Square Roots: The square root of a fraction can be simplified by taking the square root of the numerator and the denominator separately.

3. Perfect Squares: When the numerator and denominator under the square root are perfect squares (like $49=7^2$ and $25=5^2$), they can be simplified by taking the square root of each perfect square.

4. Rationalizing the Denominator: In mathematics, it's often preferred to have a rational number in the denominator rather than a radical. In this case, since the denominator is already a perfect square, it simplifies to a rational number.

5. Multiple Forms of the Result: The final result can be expressed in various forms, including an exact fraction, a decimal approximation, or a mixed number, depending on the context or preference.