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Chapter 15: Problem 52

Simplify. $$\frac{\sqrt{48}}{\sqrt{3}}$$

Short Answer

Expert verified
The simplified form of the original expression is 4.

Step by step solution

01

Rewrite the problem

The expression can be rewritten using the properties of radicals: \(\frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}}\).
02

Simplify the fraction

The fraction \(\frac{48}{3}\) simplifies to 16.
03

Take the square root

Taking the square root of 16 gives 4, so the simplified form of the expression is 4.

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Key Concepts

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

Simplifying Radicals
Simplifying radicals is all about making expressions involving roots easier to manage. By breaking down numbers under a radical, we can find simpler forms. For example, when simplifying \(\frac{\sqrt{48}}{\sqrt{3}}\), the goal is to express this in the simplest form.
  • We "combine" the radicals into one: \(\sqrt{\frac{48}{3}}\)
  • Then, simplify the fraction inside the square root: \(\frac{48}{3} = 16\)
  • Finally, take the square root of 16 which is 4.

This process turns a more complex expression into a simpler number. Remember, simplifying radicals often involves turning fractions into integers or simpler square roots.
Properties of Radicals
The properties of radicals are key to understanding how to manipulate and simplify expressions involving square roots and other roots. These properties allow us to rewrite and simplify radical expressions efficiently.
  • Product Property: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), so we can multiply under a single radical when dealing with products.
  • Quotient Property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), useful for dividing radicals, just like in our example where \(\frac{\sqrt{48}}{\sqrt{3}}\) simplifies to \(\sqrt{\frac{48}{3}}\).
  • Power Property: \(\sqrt[a]{b^c} = b^{c/a}\) helps in breaking down powers under radicals to simpler forms.

Mastering these properties makes working with radicals simpler and more straightforward. Apply them carefully, and always break down complex numbers inside the radical when possible.
Square Roots
Square roots are one of the most common types of radicals, and understanding them is essential when dealing with radical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number.
  • For instance, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).
  • Square roots of perfect squares result in integers. Perfect squares, like 1, 4, 9, 16, 25, make calculation straightforward.
  • If the number is not a perfect square, like 48 in this context, we often simplify it by breaking it into its prime factors or into perfect square components, like \(48 = 16 \times 3\).

Knowing the basics of square roots helps you recognize when numbers can be simplified under the radical and how to work out simple or more complex problems efficiently. Always check for perfect squares first, as they lead directly to simpler answers.

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Most popular questions from this chapter

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