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Chapter 10: Problem 88

Simplify. Assume that no radicands were formed by raising negative quantities to even powers. $$ \sqrt{(x+3)^{10}} $$

Short Answer

Expert verified
(x+3)^5

Step by step solution

01

Identify the Given Expression

The expression given is \ \( \sqrt{(x+3)^{10}} \). Here, we need to simplify this square root expression.
02

Use the Property of Exponents

Recall the property of exponents: \ \( \sqrt{a^b} = a^{b/2} \). Applying this property will help in simplifying the given expression.
03

Simplify the Exponent

Apply the property to the given expression: \ \( \sqrt{(x+3)^{10}} = (x+3)^{10/2} \). Simplify the fraction inside the exponent: \ \( 10/2 = 5 \). So, the expression simplifies to \ \( (x+3)^5 \).

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Key Concepts

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

Understanding Radicals
A radical, often called a root, is a way to represent the inverse operation of raising a number to a power. The most common radical is the square root, denoted \( \sqrt{} \). For example, the square root of 9 is 3, because 3\^2 equals 9.

There are also other types of roots, such as the cube root ( \( \sqrt[3]{} \) ) and the fourth root ( \( \sqrt[4]{} \) ).

Certain properties of radicals are essential for simplification:
  • \( \sqrt{a}\sqrt{b} = \sqrt{ab} \)
  • \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
  • \( \sqrt{a^2} = |a| \)
To solve \( \sqrt{(x+3)^{10}} \), we use the property that square roots can be expressed as an exponent, which we'll explore next.
The Properties of Exponents
Exponents are used to denote repeated multiplication of a number by itself. For example, \( a^n \) means a is multiplied by itself n times. Exponential properties help to simplify and solve mathematical expressions involving exponents.

Important properties of exponents include:
  • \( a^m \cdot a^n = a^{m+n} \)
  • \( \frac{a^m}{a^n} = a^{m-n} \)
  • \( (a^m)^n = a^{mn} \)
  • \( a^{0} = 1 \)
  • \( a^{-n} = \frac{1}{a^n} \)
Applying these properties can simplify complex expressions. For instance, when simplifying \( \sqrt{(x+3)^{10}} \), we can use the property \( \sqrt{a^b} = a^{b/2} \). This makes our expression \( (x+3)^{10/2} = (x+3)^5 \).
Steps of Simplification
Simplification involves breaking down a complex expression into simpler components that are easier to understand and work with. Here are the general steps:
  • Identify the given mathematical expression.
  • Search for applicable mathematical properties or rules.
  • Break down the expression using these rules.
  • Simplify each part of the expression step by step.
  • Combine the simplified parts to get the final answer.
Let's apply these steps to the given problem:

Step 1: Identify the given expression. Here it is \( \sqrt{(x+3)^{10}} \).

Step 2: Use the property of exponents. We know \( \sqrt{a^b} = a^{b/2} \). So, \( \sqrt{(x+3)^{10}} \) becomes \( (x+3)^{10/2} \).

Step 3: Simplify the exponent. We have \( 10/2 = 5 \). So, \( (x+3)^{10/2} \) simplifies to \( (x+3)^5 \).
This systematic approach makes it easier to handle and understand complex expressions.

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}} .\) Find the absolute value of each complex number. $$ |8-6 i| $$

Let \(f(x)=x^{2} .\) Find each of the following. $$f(7+\sqrt{3})$$

Simplify. $$\frac{\frac{1}{\sqrt{w}}-\sqrt{w}}{\frac{\sqrt{w}+1}{\sqrt{w}}}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[5]{a^{4} b}}{\sqrt[3]{a b}}$$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[5]{a^{3}(b-c)^{4}} \sqrt[5]{a^{7}(b-c)^{4}} $$
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