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Chapter 8: Problem 22

Write as a radical and then simplify. \(361 / 2\)

Short Answer

Expert verified
The simplified radical expression is 19.

Step by step solution

01

Understand the Fraction

The given expression is a fraction, \(361/2\). The goal is to convert this expression to a radical form, where the denominator represents the root.
02

Express as a Radical

A fraction with a denominator \(b\), like \(a/b\), can be written as a radical \(\sqrt[b]{a}\). Here \(a=361\) and \(b=2\), so we write it as \(\sqrt{361}\).
03

Simplify the Radical

Find the square root of \(361\). To simplify \(\sqrt{361}\), determine the number that squares to \(361\). This number is \(19\) because \(19^2 = 361\).

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Key Concepts

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

Radical Expressions
Radical expressions involve roots, like square roots, cube roots, and so on. They are written using a radical sign '鈭�'. The number inside the radical is called the radicand, and it is the value you are trying to find the root of. For instance, in the expression \(\sqrt{361}\), 361 is the radicand. Radical expressions can make calculations with roots look more concise and readable.
  • They often appear in algebra, geometry, and calculus.
  • Expressing numbers and variables under root symbols simplifies complex problems.
For example, when dealing with fractions as we did in the original problem, they can be converted into radical expressions to find more understandable forms. Understanding how to convert and simplify these expressions is vital in solving math problems effectively.
Simplifying Radicals
Simplifying radicals means reducing the expression under the radical to its simplest form. This involves finding the perfect power of the root (like squares for square roots) within the radicand and rewriting the expression if possible. To simplify \(\sqrt{361}\):
  • Look for a perfect square that equals 361.
  • Calculate roots to see if they result in integers.
Here, 361 is a perfect square because it equals \(19^2\). So, \(\sqrt{361} = 19\). Finding this perfect power streamlines solving problems, as you turn complex radical expressions into their most accessible forms. This skill is not just useful in testing but also in practical math applications, like geometry where right angles and triangles often generate square roots.
Fraction to Radical Conversion
Converting a fraction to a radical expression is a systematic process. When you are given a fraction, like \(\frac{a}{b}\), you can represent it as a radical \(\sqrt[b]{a}\). Here's how this works:
  • The numerator becomes the radicand.
  • The denominator indicates which root is taken.
For example, with \(\frac{361}{2}\), the number 361 becomes the radicand under a square root since the denominator is 2, forming \(\sqrt{361}\).This process is significant in mathematics, as it allows for easy manipulation of expressions and solving of equations.Recognizing how to move from fractions to radicals, and vice versa, offers greater flexibility in handling and simplifying math problems across different fields.

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Most popular questions from this chapter

Simplify. $$27 a 3 b-\sqrt{3}-38 a b 3--\sqrt{3}+a \cdot 64 b-\sqrt{3}-b \cdot a \sqrt{3}$$

Use the distance formula to calculate the distance between the given two points. $$ (-9,7) \text { and }(-8,4) $$

Use the distance formula to calculate the distance between the given two points. $$ (5,-7) \text { and }(3,-8) $$

Simplify. $$27 a 7 b 5 c 3-c-\sqrt{3}$$

Express in radical form. $$32 / 3$$
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