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Chapter 9: Problem 3

Explain how to use the quotient property of radicals to simplify \(\sqrt{\frac{4}{25}}\)

Short Answer

Expert verified
The simplified expression of \(\sqrt{\frac{4}{25}}\) is \(\frac{2}{5}\).

Step by step solution

01

Identify the Numerator and Denominator

In the radical expression \(\sqrt{\frac{4}{25}}\), the number '4' is the numerator and '25' is the denominator.
02

Apply the Quotient Property of Radicals

Next, apply the quotient property of radicals to the expression. This changes the expression from \(\sqrt{\frac{4}{25}}\) to \(\frac{\sqrt{4}}{\sqrt{25}}\).
03

Simplify the Radicals

The square root of '4' is '2' and the square root of '25' is '5'. Substituting these values into the expression results in \(\frac{2}{5}\).

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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 an essential step in making complex mathematical expressions easier to work with. Imagine you have a radical expression like \( \sqrt{\frac{4}{25}} \). Rather than trying to deal with this intimidating fraction under a single square root, our goal is to turn it into a simpler form.
  • This is where the quotient property of radicals comes into play.
  • The property lets you "split" radicals into separate easier parts.
  • Specifically, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
Breaking it down makes it more manageable, but remember鈥攚e simplify until no perfect squares are under the radical symbol. By breaking these apart into simpler radicals, you can calculate easier and avoid mistakes.
Radical Expressions
A radical expression is any mathematical expression that includes a square root, cube root, or any higher order root. Understanding how to manipulate these expressions is crucial for solving various mathematical problems.
  • In the radical expression \( \sqrt{\frac{4}{25}} \), both the numerator '4' and the denominator '25' are perfect squares.
  • This allows us to simplify them separately using their square roots.
By applying the quotient property of radicals, we express \( \sqrt{\frac{4}{25}} \) as \( \frac{\sqrt{4}}{\sqrt{25}} \), which is much easier to handle and simplifies to \( \frac{2}{5} \). It's essential to recognize when terms within the radicals are perfect squares, as this knowledge is the key to efficiently simplifying.
Square Roots
The concept of square roots is fundamental when simplifying radicals and working with radical expressions.
  • The square root of a number is one of its two equal factors, essentially the number multiplied by itself.
  • For instance, the square root of '4' is '2' because \( 2 \times 2 = 4 \).
  • Similarly, the square root of '25' is '5', as \( 5 \times 5 = 25 \).
Recognizing perfect squares within an expression allows you to replace the radical sign with its calculated value. In \( \sqrt{\frac{4}{25}} \), identifying these allows us to arrive quickly at \( \frac{2}{5} \). These roots should be memorized for efficiency in calculations, as they'll make radical simplification a breeze.

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