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Chapter 8: Problem 108

Write each expression in radical form. $$ \left(c^{2}+d^{2}\right)^{1 / 2} $$

Short Answer

Expert verified
\( \sqrt{c^2 + d^2} \)

Step by step solution

01

Understanding the Expression

We start with the expression \( \left(c^{2}+d^{2}\right)^{1 / 2} \). This expression involves a fractional exponent of \( \frac{1}{2} \), which indicates that it is written in exponent form.
02

Identifying Radical Form

The exponent \( \frac{1}{2} \) corresponds to the square root operation. Therefore, when we convert an expression from exponent form to radical form, \( a^{\frac{1}{2}} \) becomes \( \sqrt{a} \).
03

Applying Transformation

Apply the transformation to the given expression. Replace the exponent \( \frac{1}{2} \) with a square root sign: \( \sqrt{c^2 + d^2} \).
04

Final Radical Form

The radical form of the given expression \( \left(c^{2}+d^{2}\right)^{1 / 2} \) is \( \sqrt{c^2 + d^2} \).

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Key Concepts

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

Understanding Fractional Exponents
Fractional exponents are a fascinating part of algebra that can initially seem confusing, but they're not too difficult once you get the hang of them. A fractional exponent is basically another way to write a root of a number. When you see something like the exponent \( \frac{1}{2} \), it means you're dealing with a square root.
  • The numerator of a fractional exponent tells you the power you are raising your number to.
  • The denominator of a fractional exponent tells you what root to take.
  • For example, \( a^{1/2} \) is simply the square root of \( a \).
So, when you encounter a fractional exponent like \( \frac{1}{2} \), think square root! This understanding forms the basis for converting expressions from exponent form to radical form.
The Square Root Simplified
The square root is one of the most common and useful radical operations you'll encounter. It's symbolized by the radical sign \( \sqrt{} \). Intuitively, the square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \).Here's how it works:
  • \( \sqrt{x} \) is defined as the number \( y \) such that \( y^2 = x \).
  • In other words, \( y \) is the square root of \( x \).
  • The square root of a perfect square (like 4, 9, 16) is an integer.
In our exercise, \( \left(c^{2}+d^{2}\right)^{1 / 2} \) becomes \( \sqrt{c^2 + d^2} \), effectively using the idea that \( a^{1/2} \) equates to \( \sqrt{a} \).
Converting Between Exponent Form and Radical Form
Exponent form and radical form are two ways of expressing the same mathematical concept, and switching between them is a skill you'll often need.
  • Exponent form uses powers to express roots, such as \( a^{1/n} \).
  • Radical form uses the root symbol, such as \( \sqrt[n]{a} \).
To convert from exponent form to radical form:
  • Take the base of your expression in exponent form.
  • The denominator of your exponent becomes the index of the radical.
  • So \( a^{1/n} \) transforms into \( \sqrt[n]{a} \).
In your exercise, \( \left(c^{2}+d^{2}\right)^{1 / 2} \) and \( \sqrt{c^2 + d^2} \) are two expressions of the same value, highlighting how fractional exponents link deeply into radical expressions.

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Most popular questions from this chapter

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