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Chapter 8: Problem 25

Simplify each radical. $$\sqrt{27 x^{2}}$$

Short Answer

Expert verified
\(\sqrt{27x^{2}}\) simplifies to \(3\sqrt{3} * x\)

Step by step solution

01

Factorize the Radicand

First, factorize the radicand \(27x^{2}\). We can write-down \(27\) as \(3^{3}\) and \(x^{2}\) as \(x * x\). Hence, \(27x^{2} = 3^{3} * x * x\).
02

Apply the Rule of Simplification

Next, apply the rule \(\sqrt{a^{2}} = a\) for simplifying radicals. The square root of a square is the base itself. We know from Step 1 that our radicand can be expressed as \(3^{3} * x * x\). Thus \(\sqrt{27x^{2}}\) simplifies to \(3\sqrt{3} * x\).
03

Write down the final simplified form

After simplification, the given radical reduces to \(3\sqrt{3} * x\), which is the final answer.

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Key Concepts

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

Radicands
The term "radicand" refers to the number or expression inside a radical symbol, which signifies that it is under the square root. In the example \(\sqrt{27x^{2}}\), the radicand is \(27x^{2}\). Understanding how to work with radicands is a crucial step when simplifying radicals.

Here are the key things to know about radicands:
  • Factorization: Breaking down the radicand into its prime factors is the first step in simplifying radicals. This involves recognizing all the components that make up the number or expression.
  • Prime Factorization: In the example, \(27\) is factorized into \(3^3\), and \(x^2\) is written as \(x \cdot x\). When combined, it becomes \(3^3 \cdot x \cdot x\).
By restructuring the radicand into smaller parts, it becomes easier to identify pairs when applying the square root simplification rule. The focus is always to turn complex expressions into manageable pieces, preparing them for the next steps in solving square roots.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. The square root symbol \(\sqrt{}\) denotes the operation. For example, \(\sqrt{16} = 4\), since \(4 \cdot 4 = 16\). Square roots simplify square factors within expressions.
  • Paired Factors: In our exercise, the goal is to identify pairs in the factorized form of the radicand. This is based on the principle \(\sqrt{a^{2}} = a\), where \(a\) is a positive real number.
  • Simplifying Radicals: Apply the square root to all paired numbers and variables. In the exercise, the expression \(3^3 \cdot x \cdot x\) simplifies by taking \(x\) out of the square root, since \(x \cdot x\) forms a perfect square.
This process helps strip down the radical to its simplest terms, removing entire pairs from underneath the radical and converting it to more straightforward algebraic forms, like \(3\sqrt{3} \cdot x\).
Algebraic Expressions
Algebraic expressions contain numbers, variables, and operations. In the case of radicals, simplifying an expression means rewriting it in a simpler or more comprehensible form. Factors and powers play significant roles in breaking down more substantial expressions.
  • Combination of Constants and Variables: An expression like \(27x^{2}\) includes both a numerical constant (27) and a variable part \(x^{2}\). These components together make up an algebraic expression.
  • Simplification Techniques: Factoring and simplifying radical expressions are primary methods used to make algebraic expressions more accessible. Once factored as in \(3^3 \cdot x \cdot x\), we then simplify using known rules \(\sqrt{a^2} = a\) \(\text{for each pair of } a\.\)
Simplifying radicals is just one aspect of handling algebraic expressions. The technique allows the expression to be rewritten as a product of simpler terms like \(3\sqrt{3} \cdot x\), which is easier to understand and use in further calculations.

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