Problem 23 - Algebra 2

Question

Simplify. $$ \sqrt{147} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{147}\) is \(7\sqrt{3}\).

Step 1: Identify the prime factors

Start by finding the prime factors of 147. Divide 147 by the smallest prime number 3, which results in 49. Since 49 is not divisible by 3, divide 49 by 7, getting 7. Thus, the prime factorization of 147 is \(147 = 3 \times 7 \times 7\).

Step 2: Rewrite the square root expression

Using the prime factors, rewrite the square root as \(\sqrt{147} = \sqrt{3 \times 7 \times 7}\). This will help us identify perfect squares within the expression.

Step 3: Simplify by extracting perfect squares

In the expression \(\sqrt{3 \times 7 \times 7}\), \(7 \times 7 = 49\) is a perfect square. Thus, \(\sqrt{3 \times 7 \times 7} = \sqrt{3 \times 49}\). This can be simplified as \(\sqrt{49} = 7\), resulting in \(7\sqrt{3}\).

Step 4: Confirm the simplification

Verify that \(7\sqrt{3}\) is in its simplest form. There are no further simplifications as \(3\) is a prime number, confirming that the simplest form of \(\sqrt{147}\) is \(7\sqrt{3}\).

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. A prime number is a number that can only be divided by 1 and itself without leaving a remainder. Identifying the prime factors can simplify complex expressions, like square roots, making calculations more manageable.

• To start, we look for the smallest prime number that divides the number exactly.
• In our example, 147 can be divided by 3 first, resulting in 49.
• We proceed with the next step of dividing 49 by 7, yielding a neat result because 49 is a perfect square of 7.

Recognizing these factors aids in rewriting expressions, which is particularly useful for simplifying radicals. Once you know the prime factors, you can spot perfect squares more easily within the product, which prepares for further simplification of the square root.

Square Roots

A square root involves finding a number which, when multiplied by itself, gives the original number. It's like the opposite of squaring a number. Simplifying square roots often requires identifying and pulling out perfect square factors from under a radical.

• In our case, we took the square root of 147. This meant looking for any pairs of numbers that could be turned into a perfect square to simplify the expression.
• Using the expression \[\sqrt{3 \times 7 \times 7}\]we see that\[7 \times 7\]forms a perfect square.

Once identified, perfect squares allow the square root to be partially simplified, because the square root of a perfect square is a whole number. This makes expressions not only easier to compute but also more visually and computationally manageable.

Perfect Squares

Perfect squares are numbers that result from squaring integers. For example, both 1 and 49 are perfect squares of numbers like 1 and 7 respectively. Recognizing these helps to streamline otherwise complex mathematical expressions.

• In our example of simplifying \(\sqrt{147}\), we used the perfect square of 49, which is 7 squared.
• Identifying a perfect square in an expression allows it to be simplified outside the radical, reducing the complexity.

This simplification leads us to the expression \(7\sqrt{3}\), where 7 is the square root of the perfect square 49. Knowing how to identify and utilize perfect squares is a handy skill when working with square roots and other mathematical operations.