Problem 41 - Algebra 2

Question

Simplify. \(5 \sqrt{20}+\sqrt{24}-\sqrt{180}+7 \sqrt{54}\)

Step-by-Step Solution

Answer

The simplified expression is \(4\sqrt{5} + 23\sqrt{6}\).

Step 1: Break down each square root term into prime factors

First, let's break down each of the square root terms into their prime factors to simplify them. - \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\)- \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\)- \( \sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}\)- \( \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}\)

Step 2: Substitute simplified square roots back into the expression

Now substitute back the simplified square root terms into the original expression:\[5 \times 2\sqrt{5} + 2\sqrt{6} - 6\sqrt{5} + 7 \times 3\sqrt{6}\]

Step 3: Multiply and simplify the coefficients

Carry out the multiplications:- \(5 \times 2\sqrt{5} = 10\sqrt{5}\)- \(7 \times 3\sqrt{6} = 21\sqrt{6}\)So the expression simplifies to:\[10\sqrt{5} + 2\sqrt{6} - 6\sqrt{5} + 21\sqrt{6}\]

Step 4: Combine like terms

Combine the terms with similar radicals:- Combine \(10\sqrt{5} - 6\sqrt{5} = 4\sqrt{5}\)- Combine \(2\sqrt{6} + 21\sqrt{6} = 23\sqrt{6}\)The final simplified expression is:\[4\sqrt{5} + 23\sqrt{6}\]

Key Concepts

Square Roots

Square roots involve finding a number that, when multiplied by itself, gives you the original number. For example, the square root of 25 is 5, because multiplying 5 by itself (5 x 5) equals 25. Square roots are common in many areas of algebra, and in problems like the one you're working on, they can often be simplified.
To simplify square roots, it's all about breaking them down into their smallest components. This is where prime factorization comes in. Notice how \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} \), where 4 is a perfect square.

• A perfect square is a number whose square root is a whole number. In this case, the square root of 4 is 2, making part of the simplification possible.
• Breaking a number into its prime factors can help identify these perfect squares easily.

Learning to spot perfect squares and simplify them can be an invaluable skill when facing similar exercises. It makes expressions with square roots cleaner and easier to work with.

Prime Factors

Identifying prime factors is an essential skill when learning to simplify radicals like square roots. A prime number is one that is only divisible by 1 and itself, such as 2, 3, 5, 7, etc.
To simplify square roots, you break the number down into its prime factors and then look for pairs of factors. Each pair of identical factors within the square root can be pulled out as a single number, simplifying the expression.

• For instance, with \(180\): this can be broken down to \(2 \times 90\), and further into \(2 \times 2 \times 3 \times 3 \times 5\). Here, you see there are pairs of 2s and 3s.
• The equals \( \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5} \), as 36 is a perfect square.

Once you identify the perfect squares these prime factors can make, they can be taken out of the radical, leaving behind any remaining prime numbers.

Combine Like Terms

Combine like terms to simplify expressions further after the square roots are individually simplified. Like terms are terms that have the same variable raised to the same power. In the context of square roots, this means looking for common radicands (the number inside the square root) among terms.
In the exercise, you simplified everything to end up with terms like \(10 \sqrt{5}\) and \(6 \sqrt{5}\). Since both terms have the same radicand 5, you can combine them:

• Combine \(10\sqrt{5} - 6\sqrt{5} = 4\sqrt{5}\).
• The same applies to expressions like \(2\sqrt{6} + 21\sqrt{6} = 23\sqrt{6}\).

By carefully combining like terms, you make your expression as simplified as possible. This step is akin to collecting similar objects together to count them easier.