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Chapter 15: Problem 49

Simplify. $$\frac{\sqrt{32}}{\sqrt{2}}$$

Short Answer

Expert verified
The simplified form of the expression \(\frac{\sqrt{32}}{\sqrt{2}}\) is 4.

Step by step solution

01

Break Down the Square Roots

Break down the square roots into more manageable terms. The square root of 32 can be broken down to \(\sqrt{16} * \sqrt{2}\), which simplifies further to 4\(\sqrt{2}\). The square root of 2 simply remains as \(\sqrt{2}\). The expression thus becomes \(\frac{4\sqrt{2}}{\sqrt{2}}\).
02

Simplify the Expression

Divide the terms by \(\sqrt{2}\). Thus the \(\sqrt{2}\) in the numerator and the \(\sqrt{2}\) in the denominator cancel out, simply leaving the expression as 4.

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Key Concepts

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

Square Root Properties
Square roots are fundamental in mathematics, especially when simplifying expressions. They are exceptional in that they always return the principal (non-negative) square root. This means the expression \( \sqrt{x} \) represents the positive number whose square is \( x \).
  • The square root of a product is the product of the square roots: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
  • Using the property above, \( \sqrt{32} \) can be decomposed: \( 32 = 16 \times 2 \), so \( \sqrt{32} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \).
  • Square roots simplify when their radicand (the number inside the root) is a perfect square, e.g., \( \sqrt{9} = 3 \).
By understanding and applying these properties appropriately, complex square root expressions become more manageable. This concept lays the groundwork for further explorations in algebra and beyond.
Fraction Simplification
Simplifying fractions involves reducing them to their simplest form. This is a key step in algebra to make expressions more interpretable.
  • Consider the fraction \( \frac{\sqrt{32}}{\sqrt{2}} \). The aim is to simplify both the numerator and the denominator.
  • Once simplified to \( \frac{4\sqrt{2}}{\sqrt{2}} \), the identical terms in both the numerator and the denominator can be cancelled out.
  • Canceling \( \sqrt{2} \) from both parts, the fraction simplifies to 4.
It is crucial to understand that canceling terms this way only works because both parts are multiplied or divided by the same non-zero terms. This step is particularly helpful in keeping the arithmetic part of algebraic expressions clean and concise.
Prealgebra Concepts
Prealgebra serves as the foundation for learning algebra and higher-level math. It introduces key concepts and operations that become building blocks for solving complex problems.
  • The understanding of operations such as division, multiplication, addition, and subtraction is developed in this stage.
  • Simplification skills, like reducing fractions and working with square roots, are crucial. These skills ensure you can handle equations more efficiently.
  • Prealgebra aims to prepare you for algebraic thinking by making you comfortable with numerical expressions and their manipulations.
Mastering prealgebra concepts greatly enhances your problem-solving ability and sets a solid groundwork for future mathematical success.

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