Problem 18 - Algebra 2

Question

Simplify. \(\sqrt{243}\)

Step-by-Step Solution

Answer

\(\sqrt{243} = 9\sqrt{3}\).

Step 1: Prime Factorization

To simplify the square root of 243, we first need to find its prime factors. Divide 243 by 3: 243 ÷ 3 = 81. Then 81 by 3: 81 ÷ 3 = 27. Continue with 27: 27 ÷ 3 = 9. 9 by 3: 9 ÷ 3 = 3. Finally, 3 by 3: 3 ÷ 3 = 1. So, 243 = 3^5.

Step 2: Group Prime Factors

We need to identify pairs of identical factors since we are dealing with a square root. From the factorization in step 1, we have 3^5 = 3×3×3×3×3. Group these into pairs: (3×3) and (3×3), with one 3 left over.

Step 3: Apply the Square Root Property

The property of square roots allows us to take one factor from each pair outside of the square root sign. Therefore, \(\sqrt{243} = \sqrt{(3×3)×(3×3)×3} = 3×3×\sqrt{3} = 9\sqrt{3}\).Thus, the simplified form is 9\sqrt{3}.

Key Concepts

Prime Factorization

To simplify a square root such as \(\sqrt{243}\), the first step is to perform prime factorization. This involves breaking down a number into its smallest building blocks—prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. By factorizing 243, we start by dividing it by the smallest prime number, which is 3 in this case. The division process would look like this:

• 243 divided by 3 equals 81
• 81 divided by 3 equals 27
• 27 divided by 3 equals 9
• 9 divided by 3 equals 3
• Finally, 3 divided by 3 equals 1

By repeatedly dividing by 3, we determine that the complete factorization of 243 is \(3^5\), or five 3s multiplied together: \(3\times3\times3\times3\times3\). Prime factorization is crucial for simplifying radicals as it identifies the building blocks that can be paired and simplified further.

Square Root

Once we have the prime factorization of a number, we can move on to simplifying its square root. A square root, denoted by the symbol \(\sqrt{}\), of a number is a value that, when multiplied by itself, gives the original number. When simplifying \(\sqrt{243}\), we utilize the prime factors we discovered: \(3^5\). Each pair of identical numbers under a square root can be taken "outside" as a single number. For \(3^5\), we have two pairs of 3s and one 3 remaining single.

• Two pairs: (3 \(\times\) 3) and (3 \(\times\) 3)
• One leftover 3

Therefore, when applying the square root, each pair of 3s becomes a single 3 outside the square root, resulting in: \(3 \times 3 \times \sqrt{3} = 9\sqrt{3}\). This expression is the simplified form of the original square root.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. In mathematics, an exponent refers to the number of times a number, known as the base, is multiplied by itself. In our problem, the number 3 was raised to the power of 5, written as \(3^5\). This notation means \(3\times3\times3\times3\times3\).
When simplifying square roots, understanding exponents is important because it helps us form groups of pairs needed for taking a number out of the square root sign.
In the case of \(3^5\), we restructured it as \((3\times3)\times(3\times3)\times3\). Recognizing these pairs makes it easier to apply the square root rules.

Knowing how to work with exponents not only aids in simplifying radicals but also in various other areas of math, providing insight into how numbers interact in equations and expressions.