Question

Simplify the expression. $$\frac{\sqrt{36}}{\sqrt{9}}$$

Short answer

The simplified form of the expression \( \frac{\sqrt{36}}{\sqrt{9}} \) is 2

Step-by-step solution

Calculate the square root of the numerator

Calculate the square root of 36. The square root of 36 is 6 because 6 * 6 = 36.

Calculate the square root of the denominator

Calculate the square root of 9. The square root of 9 is 3 because 3 * 3 = 9.

Simplify the fraction

Now we have a fraction where the numerator is 6 and the denominator is 3. Simplify the fraction by dividing the numerator by the denominator, giving a result of 2.

Key concepts

Square Roots

Understanding square roots is essential when it comes to simplifying radical expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, when you're asked to find \(\sqrt{36}\), you're looking for a number which, when multiplied by itself, equals 36. Thus, \(\sqrt{36} = 6\) because \(6 \times 6 = 36\). It's like asking, what number doubled in area gives me this total area? Square roots are like the reverse operation of squaring a number.

Keep in mind the principle that if \(a^2 = b\), then \(a\) is the square root of \(b\). This reverse operation is crucial while solving a wide range of mathematical problems, from basic algebra to more complex calculus problems. Additionally, identifying perfect squares, which are integers that are squares of other integers (like 9, 16, 25, etc.), can make the calculation of square roots much simpler in many cases.

Simplifying Fractions

When dealing with fractional expressions, simplifying fractions is a critical skill. Simplifying a fraction means transforming it into its simplest form where the numerator and denominator are as small as possible, but still have the same value. This is done by dividing both the numerator and the denominator by their greatest common factor (GCF).

Let's take the fraction \(\frac{6}{3}\) as an example. Here, the GCF of 6 and 3 is 3. By dividing both by 3, you get \(\frac{6 \div 3}{3 \div 3} = \frac{2}{1}\), which is simply 2. You'll often encounter this step in algebra, particularly when working with expressions that involve radicals. Keeping your fractions simplified can help you spot patterns and make further calculations much easier.

Arithmetic Operations

Arithmetic operations are the foundation of mathematics, which include addition, subtraction, multiplication, and division. In the context of simplifying radical expressions with fractions, multiplication and division are the key players. These basic operations enable us to combine or break apart numbers and expressions in a systematic way.

Multiplying and dividing roots can be particularly interesting. When multiplying roots, you multiply the numbers inside the radicals. As for division, it entails dividing the numbers inside the roots, given that the roots are of the same index. For example, when you're given \(\frac{\sqrt{36}}{\sqrt{9}}\), you are essentially performing the operation of division between two square roots, which is simplified by dividing the numbers under the radicals and taking the root of the quotient.

Radical Expressions

Radical expressions involve numbers under the radical sign. Simplifying these expressions often requires applying the rules for square roots and other arithmetic operations. A radical expression can include numbers, variables, or both under the radical sign. Simplification may involve finding perfect square factors and separating them from the radical. In doing so, you transform the expression into a simpler form without changing its value.

For instance, the expression \(\frac{\sqrt{36}}{\sqrt{9}}\) includes radicals in both the numerator and the denominator. Simplifying this radical expression means applying the square root to both numbers and then dividing them as per the regular arithmetic rules. The aim is always to express the radicals in their simplest form to make mathematical operations easier and more intuitive.