Problem 101
Question
Simplify. $$ \sqrt{\frac{49}{64}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \( \frac{7}{8} \).
1Step 1: Recognize the Radical Expression
Identify that the expression is a square root of a fraction: \ \ \( \sqrt{\frac{49}{64}} \ \)
2Step 2: Simplify the Numerator and Denominator Separately
Take the square root of the numerator (49) and the denominator (64) separately: \ \ \( \sqrt{49} = 7 \) \ \ \( \sqrt{64} = 8 \)
3Step 3: Write the Simplified Expression
Combine the simplified parts: \ \ \( \frac{7}{8} \)
Key Concepts
Square RootsFractionsBasic Algebra
Square Roots
Understanding square roots is crucial in simplifying radical expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 49 is 7 because 7 multiplied by 7 equals 49. Similarly, the square root of 64 is 8 because 8 multiplied by 8 equals 64.
In mathematical notation, we write this as \[ \sqrt{49} = 7 \quad \text{and} \quad \sqrt{64} = 8 \].
The concept of square roots can be extended to fractions. For a fraction under a square root, we can simplify it by taking the square root of the numerator and the denominator separately.
In mathematical notation, we write this as \[ \sqrt{49} = 7 \quad \text{and} \quad \sqrt{64} = 8 \].
The concept of square roots can be extended to fractions. For a fraction under a square root, we can simplify it by taking the square root of the numerator and the denominator separately.
Fractions
Fractions are parts of a whole expressed in terms of a numerator (top part) and a denominator (bottom part). In our exercise, we are dealing with the fraction \( \frac{49}{64} \). To simplify this within a square root, we take the square root of both the numerator and the denominator individually.
This approach works because of the property of square roots that states: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
Applying this, we get: \( \sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8} \).
After simplifying, \( \frac{7}{8} \) is in its simplest form where both the numerator and denominator are whole numbers and have no common factors other than 1.
This approach works because of the property of square roots that states: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
Applying this, we get: \( \sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8} \).
After simplifying, \( \frac{7}{8} \) is in its simplest form where both the numerator and denominator are whole numbers and have no common factors other than 1.
Basic Algebra
Basic algebra involves various operations and manipulations of mathematical expressions. In our example, we applied the concept of simplifying a square root within a fraction. This is a fundamental algebraic manipulation and relies on knowing properties of square roots and fractions.
Here's a quick recap of the steps:
With these steps, we transformed \( \sqrt{\frac{49}{64}} \) into \( \frac{7}{8} \). Understanding these principles helps to tackle more complex algebraic problems in the future.
Here's a quick recap of the steps:
- Identify the square root of a fraction.
- Simplify the numerator and the denominator separately by taking their square roots.
- Combine the simplified results into one fraction.
With these steps, we transformed \( \sqrt{\frac{49}{64}} \) into \( \frac{7}{8} \). Understanding these principles helps to tackle more complex algebraic problems in the future.