Problem 27 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{160}\)

Step-by-Step Solution

Answer

\(\sqrt{160} = 4\sqrt{10}\).

Step 1: Identify Factors of 160

Start by finding the prime factorization of the number under the square root, which is 160. Begin by dividing 160 by the smallest prime number, which is 2, and continue until all factors are prime. 160 ÷ 2 = 80 80 ÷ 2 = 40 40 ÷ 2 = 20 20 ÷ 2 = 10 10 ÷ 2 = 5 5 is a prime number. Therefore, the prime factorization of 160 is 2 × 2 × 2 × 2 × 2 × 5 or using exponents: 2^5 × 5.

Step 2: Simplify the Radical

Now, rewrite the square root expression using the prime factorization:\[ \sqrt{160} = \sqrt{2^5 \times 5} \]Pair the factors under the square root (because the square root of a² is a):Since we have 2⁵, it can be grouped as (2²)² × 2² × 2 × 5.Take out the pairs:\[ \sqrt{(2^2)^2 \times 2 \times 5} = \sqrt{(4)^2 \times 2 \times 5} \]The square root of \((4)^2\) is 4, so take 4 out of the radical:\[ \sqrt{160} = 4\sqrt{10} \]

Step 3: Write the Simplified Answer

The expression \(\sqrt{160}\) in simplest radical form is expressed as 4\(\sqrt{10}\).

Key Concepts

Prime Factorization

Prime factorization is a method used to simplify numbers by breaking them down into their basic building blocks: prime numbers. A prime number is a number that only has two distinct positive divisors: 1 and itself. For example, 2, 3, and 5 are all prime numbers.
When we talk about prime factorization, we are interested in representing a number as the product of these prime numbers. In the case of simplifying radicals, like \(\sqrt{160}\), prime factorization helps us break down the number inside the square root to a simpler form.

• Start by identifying the smallest prime number, which is 2.
• Keep dividing the number by 2 until it is no longer divisible by 2.
• Once 2 is no longer a factor, move on to the next smallest prime number and continue this process.

Using the example of 160, we divide by 2 repeatedly until we're left with 5, a prime number. Thus, 160 can be factorized as \(2^5 \times 5\). This factorization is useful for pairing factors when simplifying the square root.

Square Root

Understanding the square root is essential for simplifying radicals. The square root of a number \(n\) is a value that, when multiplied by itself, equals \(n\). Essentially, it "undoes" squaring a number. Using mathematical notation, this is written as \(\sqrt{n}\).
In the context of radical simplification, the square root symbol denotes that we're looking to "break down" the number into more manageable factors. This process is simplified immensely by using prime factorization.When you deal with a radical like \(\sqrt{160}\), the prime factorization of the number, \(2^5 \times 5\), helps us identify pairs of numbers. Since the square root of a pair of identical numbers is the number itself (for example, \((a^2)^{1/2} = a\)), you can "extract" these pairs from under the square root symbol as single numbers.
This concept is crucial for correctly simplifying expressions that involve square roots. By applying this rule, you can turn a complex radical into a much simpler form.

Simplest Radical Form

The simplest radical form is a way of expressing a radical such that there are no perfect squares left under the square root.This form helps to "clean up" a radical expression, making it easier to work with for further calculations.

• To achieve the simplest radical form, we start by performing the prime factorization.
• Next, we pair the factors under the square root symbol.
• Remove the pairs from the radical, converting them into single numbers outside the radical symbol.

For the example \(\sqrt{160}\), after factoring into \(2^5 \times 5\), we identify that there are pairs of 2's that can come out from under the square root: \(\sqrt{(4)^2 \times 2 \times 5} = 4\sqrt{10}\).
This is the simplest radical form of \(\sqrt{160}\), as it no longer contains squares under the radical, simplifying calculations and providing a clear, concise representation of the original expression.