Simplify square root of 1 24/25
The given problem is asking for the simplification of the square root of an improper fractional number. The improper fraction in question is 1 24/25, which means there is a whole number part (1) and a fractional part (24/25). The task is to combine these parts into a single fractional value, convert this into an equivalent radical expression, and then simplify the square root of that expression. Simplifying here usually means finding the square root in radical form, in simplest form, or as a decimal if possible.
$\sqrt{1 \frac{24}{25}}$
Answer
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:
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.
Simplifying Square Roots: The square root of a fraction can be simplified by taking the square root of the numerator and the denominator separately.
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.
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.
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.
