Problem 17 - Algebra 1

Question

Simplify the expression. $$\sqrt{80}$$

Step-by-Step Solution

Answer

The simplified expression is \(4\sqrt{5}\).

Step 1: Find the Factors

Start by factoring 80 into its prime factors. The prime factors of 80 are 2, 2, 2, 2, and 5

Step 2: Identify the Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. In the prime factors of 80, we have four twos. Pair them in twos to form perfect squares i.e. \((2*2 = 4)\). Now replace these perfect squares in the original square root expression.

Step 3: Simplify the Square Root

Take the square root of the perfect squares and multiply the result with the square root of the remaining factors. So, we have \( \sqrt{4} * \sqrt{4} * \sqrt{5} = 2 * 2 * \sqrt{5}\)

Step 4: Final Simplification

Finally, multiply the outside numbers together and keep it separate from the remaining square root. This gives us our simplified expression, i.e. \(4\sqrt{5}\)

Key Concepts

Understanding Prime Factorization

Prime factorization is a method used to break down a whole number into its fundamental building blocks, which are prime numbers. A prime number is a number greater than 1 that cannot be divided evenly by any other numbers except for 1 and itself. To find the prime factorization of a number, we repeatedly divide the number by its smallest prime factor, starting from 2, until we reach 1.

• For example, to factorize 80, we start with the smallest prime, 2. Since 80 is divisible by 2, we divide it to get 40. We continue dividing by 2 until it no longer fits, which leads us eventually to have 2, 2, 2, 2, and 5 as prime factors.


• This approach breaks down 80 as 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 5. Each of these numbers is prime, confirming we have the complete set of prime factors for 80.

This method is beneficial in simplifying square roots and other operations since it reveals all the component parts of the number in question.

Perfect Squares and Their Role in Simplifying Expressions

A perfect square is an important concept in mathematics. It refers to a number that is the result of an integer multiplied by itself. Examples include 4 (since 2 \( \times \) 2 = 4) and 9 (since 3 \( \times \) 3 = 9).

• In the case of simplifying the square root of 80, identifying perfect squares among its factors helps simplify the expression. From the prime factorization of 80, we derived 2, 2, 2, 2, and 5. Among these, 2 \( \times \) 2 and another 2 \( \times \) 2 form perfect squares, both resulting in 4.


• We can replace these pairs of twos with the result of their multiplication, allowing us to transform the expression \( \sqrt{80} \) into \( \sqrt{4 \times 4 \times 5} \).

By pulling out those perfect squares, this step makes it easier to compute, as perfect squares have simple square roots.

Square Roots: Simplifying Complex Expressions

Square roots are often used to simplify expressions involving powers. The square root of a number, \( x \), is a value that, when multiplied by itself, gives \( x \). Knowing how to handle square roots can reveal simpler forms of complex expressions.

• In mathematical terms, \( \sqrt{a} \) represents the number whose square equals \( a \). Consider the expression \( \sqrt{80} \). By identifying perfect squares within the prime factorization (4 \( \times \) 4 \( \times \) 5), we know \( \sqrt{4} \) equals 2.


• This allows us to further simplify the expression: \( 2 \times 2 \times \sqrt{5} \), which results in 4\( \sqrt{5} \).

This method of simplification leverages known square roots and straightforward multiplication, helping to reduce long and complicated expressions into more manageable forms. Understanding and practicing these processes makes working with square roots more intuitive and less challenging.