Problem 68 - Algebra 1

Question

Simplify the radical expression. $$\sqrt{9900}$$

Step-by-Step Solution

Answer

Therefore, the simplified form of \( \sqrt{9900} \) is 2 * 5 * 11 = 110.

Step 1: Prime factorize the number

To simplify a radical, begin by prime factorizing the number under the root. In this case, prime factorize 9900. This can be done by finding numbers that multiply together to give 9900 and that cannot be further subdivided. Here, it factorizes to \(2^2 * 3^1 * 5^2 * 11^2\).

Step 2: Identify and take out pairs

A pair of factors can be removed from under the radical by taking one of the numbers of the pair out. So, identify all pairs in the prime factorisation. Here 2,5 and 11 are the pairs of factors, each appearing twice (a pair).

Step 3: Pull out the pairs

Pull out the pairs. So, take the numbers 2, 5 and 11 out of the root, which leaves nothing under the root since there are no remaining factors. This simplification is possible since the square root of a pair of identical factors equals just one of those factors.

Key Concepts

Prime Factorization

Prime factorization is like breaking down a number into its building blocks. Just like you can break a Lego house into individual Lego pieces, you can break a number into smaller numbers that multiply together to make it. These smaller numbers are called prime numbers, and they can't be split further.
For example, in the problem of simplifying \(\sqrt{9900}\), the prime factorization of 9900 gives us \(2^2 \times 3^1 \times 5^2 \times 11^2\). Each of these numbers—2, 3, 5, and 11—are prime, meaning they are only divisible by 1 and themselves.
Prime factorization helps us simplify radicals by letting us see the natural building blocks of the number. By organizing the base numbers and their exponents, we can better manipulate them to simplify expressions under a square root or other radicals.

Square Roots

The square root is like asking "what number can I multiply by itself to reach this original number?" It's a common question in math, especially when dealing with areas and exponents.
When simplifying square roots, we often look for perfect squares. A perfect square is a number that can be expressed as the product of an integer with itself. For example, the square root of 25 is 5, because \(5 \times 5 = 25\).
In the context of \(\sqrt{9900}\), we looked at the factors \(2^2\), \(5^2\), and \(11^2\). These are considered perfect squares because we're using the factors in pairs. By extracting one number from each pair (such as taking out one 2 from \(2^2\)), we simplify the number under the square root, making calculations easier.

Radical Expressions

Radical expressions include square roots, cube roots, and higher roots. These expressions can appear daunting, but they become manageable when you break them down through simplification techniques.
The core goal of simplifying radical expressions is to make them easier to work with by reducing them to their simplest forms. This often involves dealing with perfect squares or cubes, as in the prime factorization and square root sections.
In \(\sqrt{9900}\), simplifying the expression means identifying and extracting pairs of factors. When you pull out numbers such as 2, 5, and 11 from the radical, you're left with \(2 \times 5 \times 11 = 110\). This process transforms a complex radical into a simpler figure that is straightforward to handle, showing the number's elegance beneath its radical sign.