Problem 67 - Algebra 2

Question

Simplify each expression. $$ 3 \sqrt{63}+\sqrt{28} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(11\sqrt{7}\)

Step 1: Prime Factorization of the Numbers Under the Square Roots

The first step is to perform prime factorization on the numbers under the square roots. Prime factorization of \(63\) is \(3^2*7\) and prime factorization of \(28\) is \(2^2*7\). So, the expression can be written as: \(3 \sqrt{3^2*7}+\sqrt{2^2*7} \)

Step 2: Simplify Square Roots

Using the property of square roots, \(\sqrt{a^2*b} = a\sqrt{b}\), simplify \(\sqrt{63}\) and \(\sqrt{28}\). This gives \(3*3\sqrt{7} + 2\sqrt{7}\), which further simplifies to \(9\sqrt{7} + 2\sqrt{7} \). So, the expression is now \(9\sqrt{7} + 2\sqrt{7}\)

Step 3: Combine Like Terms

The last step is to combine like terms. These are terms under the same square root, in this case \(\sqrt{7}\). This becomes \( (9+2)\sqrt{7} = 11\sqrt{7} \).

Key Concepts

Prime Factorization

Prime factorization is a method used to break a number down into its smallest prime numbers. It's a bit like finding out what smaller numbers multiply together to make the original number. In the context of simplifying radicals, understanding prime factorization helps us manipulate the numbers under a square root more easily.

• Take the number 63. To factor it, start by using the smallest prime number, which is 2. Since 63 is odd, move on to 3. 63 divided by 3 gives 21. 21 divided by 3 gives 7, which is a prime number. Thus, the prime factorization of 63 is expressed as \(3^2 \times 7\).
• Now, let's factor 28. 28 is even, so divide by 2 to get 14. Divide by 2 again to get 7, which is prime. So, the prime factorization of 28 is \(2^2 \times 7\).

Using prime factorization, you can simplify the expression containing square roots, making the subsequent steps smoother. Remember, finding and correctly identifying each prime factor can drastically clarify the simplification process.

Square Roots

When simplifying square roots, you are often tasked with turning the expression under the root into something more concise. This is done by taking pairs of prime factors out of the square root.

• Apply the property \( \sqrt{a^2 \times b} = a \sqrt{b} \). If you have a square under the root, you can pull it out as a single number. For \( \sqrt{63} \), prime factorization shows \( \sqrt{3^2 \times 7} \). The square 3 comes out as 3, making it \(3 \times \sqrt{7} \).
• For \( \sqrt{28} \), prime factorization results in \( \sqrt{2^2 \times 7} \). Here, 2 escapes the root because \( \sqrt{2^2} = 2 \), giving \(2 \sqrt{7} \).

Notice how simplifying square roots using their prime factors takes complex expressions and makes them easier to work with, leading up to the final simplification step.

Combining Like Terms

Combining like terms is a straightforward yet vital step in many algebraic processes. It simply means to bring together terms that have the same variable part. For radicals, you're looking for terms with the same part under the square root.

• In our example, after simplifying the roots, we end up with \(9 \sqrt{7} + 2 \sqrt{7} \). Both terms share the \(\sqrt{7}\), making them like terms.
• To combine them, simply add up the coefficients (the numbers in front of \(\sqrt{7}\)). Here, it's 9 plus 2, which gives 11. So, \((9 + 2) \sqrt{7} = 11 \sqrt{7}\).

Combining like terms is the last piece of the puzzle in simplifying radical expressions. It reduces the expression to its simplest form, making it much tidier and more mathematically elegant.