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Chapter 0: Problem 38

Add or subtract terms whenever possible. $$ \sqrt{20}+6 \sqrt{5} $$

Short Answer

Expert verified
The simplified form of the expression \(\sqrt{20} + 6\sqrt{5}\) is \(8\sqrt{5}\).

Step by step solution

01

Simplify the square root

\(\sqrt{20}\) can be simplified by breaking it down into \(\sqrt{4 * 5}\) which is equivalent to \(2\sqrt{5}\). The expression now reads: \(2\sqrt{5} + 6\sqrt{5}\).
02

Combine like terms

Now that we have the expression \(2\sqrt{5} + 6\sqrt{5}\), we can combine like terms, similar to how one might combine 'x' terms in an algebraic equation. We add the 2 and the 6 together to get \(8\sqrt{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Square Roots
When working with square roots, it's common to encounter more complex numbers that can be simplified. Simplifying makes the expressions neater and easier to handle. The process involves simplifying the number under the square root (the radicand) by identifying any perfect square factors it might have.

For example, the square root of 20 can be simplified as follows:
  • Break down 20 into its prime factors: 20 = 4 脳 5.
  • Identify the perfect square among these factors, which is 4.
  • The square root of 4 is 2. Therefore, \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \).
Simplifying square roots is crucial, especially in algebraic equations, where it helps to combine like terms or solve equations more easily. Always look for perfect square factors in the radicand to simplify square roots efficiently.
Combining Like Terms
Once you simplify the square roots, the next step often involves combining like terms. Combining like terms is a fundamental skill in algebra that consolidates similar components in an expression.

The process of combining like terms in the previous example involves adding terms that have the same variable part. Here, both terms include \( \sqrt{5} \):
  • After simplifying, the expression becomes \( 2\sqrt{5} + 6\sqrt{5} \).
  • Both terms have \( \sqrt{5} \), so we can add the coefficients (2 and 6) together.
  • This results in \( 8\sqrt{5} \), the simplified and combined expression.
Remember to focus on the coefficients when combining like terms, and make sure that the radicals are identical to combine them. If the radicals differ, they cannot be combined directly.
Square Root Properties
Understanding square root properties can greatly simplify algebra problems. These properties allow you to manipulate square roots efficiently and are useful for both simplifying and combining terms.

Key properties include:
  • Product property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This helps in breaking down a radicand into smaller parts.
  • Simplification through perfect squares: If a radicand includes a perfect square, its root can be simplified outside the radical.
  • Like terms in square roots: You can only combine terms under a square root if the radicals are identical. This is like combining terms in algebra 鈥� they must have the same variable or radical part.
Grasping these properties gives you the tools to simplify expressions efficiently. They lay the foundation for solving larger and more complex algebraic equations that involve square roots.

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