Problem 57 - Algebra 2

Question

What is an expression for \(\sqrt{20}-\sqrt{80}+\sqrt{125} ?\) $$\begin{array}{llll}{\text { F. } \sqrt{65}} & {\text { G. } 13 \sqrt{5}} & {\text { H. } 11 \sqrt{5}} & {\text { J. } 3 \sqrt{5}}\end{array}$$

Step-by-Step Solution

Answer

The simplified expression is \(3\sqrt{5}\), making the correct answer choice J.

Step 1: Simplify \(\sqrt{20}\)

First, break 20 into its prime factors to simplify the square root. The prime factorisation of 20 is \(2^2 * 5^1\). This allows us to simplify \(\sqrt{20}\) to \(2\sqrt{5}\).

Step 2: Simplify \(\sqrt{80}\)

Breaking 80 into its prime factors gives \(2^4 * 5^1\). Therefore, \(\sqrt{80}\) simplifies to \(4\sqrt{5}\).

Step 3: Simplify \(\sqrt{125}\)

The prime factorisation of 125 is \(5^3\). So, \(\sqrt{125}\) simplifies to \(5\sqrt{5}\).

Step 4: Combine the simplified terms

After simplifying, the expression becomes \(2\sqrt{5} - 4\sqrt{5} + 5\sqrt{5}\), which can be combined to \(3\sqrt{5}\).

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a whole number as a product of prime numbers. This is a crucial step when simplifying radicals, as it helps in determining perfect squares within a number.

• To find the prime factorization, start by dividing the number by the smallest prime number, which is 2, and continue dividing by prime numbers until the quotient is 1.
• For example, the prime factorization of 20 is done by splitting 20 as 2 multiplied by 10, and then 10 is split into 2 and 5. Thus, 20 becomes \(2^2 \times 5^1\).
• The number 80 can be broken down starting with 2, since 80 is even, continuing with 40 (which becomes 20, then 10, and finally 5). Therefore, 80 becomes \(2^4 \times 5^1\).
• Similarly, 125 is expressed as \(5^3\) because 125 is 5 multiplied by 25, and 25 is 5 multiplied by 5.

Prime factorization makes it easier to identify squares within radicals, which allows for further simplification.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. Simplifying square roots often involves finding and factoring out perfect squares.

• After finding the prime factors of a number, pair the factors to find perfect squares.


• For instance, with 20, the factor \(2^2\) is a perfect square. So, \(\sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5}\).
• With 80, you find \(2^4\), a perfect square as \(4^2\), so \(\sqrt{80} = \sqrt{(2^2)^2 \times 5} = 4\sqrt{5}\).


• For 125, the factor \(5^2\) is a perfect square, resulting in \(\sqrt{125} = \sqrt{5^2 \times 5} = 5\sqrt{5}\).

Remember, identify and extract perfect square factors from under the radical sign for simplification.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. It can often involve combining like terms after simplifying components like radicals.

• Once radicals are simplified, combine them if they share the same base under the square root.


• From the example, after simplifying each radical, we have \(2\sqrt{5} - 4\sqrt{5} + 5\sqrt{5}\).
• This process is like combining like terms in algebra, where you add or subtract coefficients of terms that have the same variable.


• Perform arithmetic on the coefficients: \(2 - 4 + 5\) gives 3, keeping the \(\sqrt{5}\) term as it is. The expression is hence \(3\sqrt{5}\).

Simplifying helps make expressions more manageable and easier to interpret in calculations.