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Chapter 8: Problem 13

Determine whether the terms contain like radicals. $$2 \sqrt{3 x} \text { and } 3 \sqrt{2 x}$$

Short Answer

Expert verified
No, the terms \(2\sqrt{3x}\) and \(3\sqrt{2x}\) do not contain like radicals.

Step by step solution

01

Analyze the radicands

Looking at the radicals, we can see that one contains \(\sqrt{3x}\) and the other contains \(\sqrt{2x}\). These radicands are not the same.
02

Conclude whether the terms contain like radicals

Since the radicands of the two radicals are not the same, we can conclude that the terms do not contain like radicals.

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Key Concepts

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

Understanding Radicands
A radicand is the number or expression inside the radical symbol, also known as the square root sign (\(\sqrt{}\)). It is the term inside that bracket-like structure. In the given exercise, we have two radicands which are \(3x\) and \(2x\). These are the main parts we need to inspect to decide if we have like radicals.
  • Same Radicands: If the radicands are identical, then the radical expressions are considered "like radicals". For example, \(\sqrt{5}\) and \(7\sqrt{5}\) have the same radicand \(5\), so they are like radicals.
  • Different Radicands: If the radicands are different, even if they look similar, the radical expressions are not like radicals. For our exercise, since \(3x\) and \(2x\) are different, the radicals are not like.

    • When analyzing the radicands, it is crucial to look beyond mere numbers. If variables are involved, ensure that both the coefficients and the variable parts match completely.
    Demystifying Radicals
    Radicals are expressions that include a square root, cube root, or any higher order root. They are placed under the radical symbol (\(\sqrt{}\) for square roots). In algebra, radicals can often simplify complex expressions and make irrational numbers more manageable.
    • Square Roots: The most common radicals are square roots, represented by \(\sqrt{}\). For instance, \(\sqrt{16}\) equals 4. In our problem, both terms had square roots which result in irrational numbers unless they simplify perfectly.
    • Modifying Expressions: Radicals change how expressions behave. Multiplying or adding radicals usually requires them to be simplified first. This can make working with variables and equations much more straightforward, granting clearer results.
    When working with radicals, remember that if their radicands are the same, it's possible to add or subtract the expressions directly, similar to like terms in regular algebraic contexts.
    Algebraic Expressions Deep Dive
    Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. Radicals are often part of these expressions, giving them unique properties.
    • Variables and Coefficients: In an expression like \(2\sqrt{3x}\), 2 is the coefficient, while \(\sqrt{3x}\) is the radical part. Together, they form a term inside a larger expression.
    • Simplifying Expressions: Simplifying these expressions involves actions like combining like terms or factoring, but when radicals are involved, additional steps are necessary. The radicand must often be simplified to easily observe and handle similar terms.
    • Properties: Algebraic expressions follow specific rules, such as the distributive, associative, and commutative properties. While radicals complicate these properties slightly, they still apply as long as like radicals can be combined.
    Understanding how to manipulate algebraic expressions with radicals is a valuable skill, as it leads to better problem-solving efficiency and clarity in mathematics.

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    Most popular questions from this chapter

    If \(x>y,\) which is the larger number in each pair? $$2^{x}, 2^{y}$$

    Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$(-8)^{-4 / 3}$$

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    Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\frac{x^{4 / 5} x^{1 / 3}}{x^{2 / 15}}$$

    Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$81^{3 / 4}$$
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