Question

Simplify the expression. $$\frac{\sqrt{49}}{\sqrt{4}}$$

Short answer

The simplified expression is 3.5

Step-by-step solution

Take the square root of the numerator

In this case, the number under the square root sign in the numerator of the fraction is 49. The square root of 49 is 7, because 7 times 7 equals 49.

Take the square root of the denominator

Next, it's important to take the square root of the number in the denominator of the fraction. Here, the number under the square root sign is 4. The square root of 4 is 2, because 2 times 2 equals 4.

Divide the results

Now that we have the square root of both the numerator and denominator, we need to divide these two numbers. Taking the square root of the numerator (7) and dividing it by the square root of the denominator (2) results in a value of 3.5

Key concepts

Radical Expressions

Radical expressions are mathematical expressions that contain a square root, cube root, or any higher order root. They play a fundamental role in algebra and appear frequently in various mathematical contexts, including geometry, calculus, and physics. One of the most common types of radical expressions is the square root, which is used to determine a number that, when multiplied by itself, gives the original number. For example, in the expression \( \sqrt{49} \), the number 7 is the square root of 49 because \( 7 \times 7 = 49 \).

When simplifying radical expressions, the goal is to find the simplest form of the expression. This often involves identifying perfect squares, which are numbers that have an integer square root, as seen in the given exercise with 49 and 4. Simplification can also involve operations with variables under the radical sign and using the properties of exponents to express the radicals in a more straightforward manner.

Square Root Operations

Square root operations involve finding the root of a number or performing arithmetic with square roots. In the context of the given exercise, these operations entail taking the square root of both the numerator and the denominator.

To perform these operations accurately, one must remember that the square root of a number is the value that, when squared, returns to the original number. For instance, \( \sqrt{4} \) simplifies to 2 because \( 2^2 = 4 \). Performing division with square roots, as needed in this exercise, involves dividing the square root of the numerator by the square root of the denominator, resulting in \( \frac{\sqrt{49}}{\sqrt{4}} = \frac{7}{2} \), which simplifies to 3.5.

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate square roots from the denominator of a fraction. Although not required in the given exercise, this concept is important to understand for working with more complex radical expressions. To rationalize a denominator, one multiplies the fraction by a form of 1 that will eliminate the square root in the denominator. This form of 1 usually contains the radical that is in the denominator.

For example, if we had a fraction like \( \frac{5}{\sqrt{2}} \), we would rationalize the denominator by multiplying both the numerator and the denominator by \( \sqrt{2} \), resulting in \( \frac{5\sqrt{2}}{2} \). This process ensures that the denominator is a rational number, which can be more convenient for further mathematical operations.