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Chapter 7: Problem 103

Simplify. Assume that \(x \geq 0 .\) \(\sqrt[4]{48^{2}}\)

Short Answer

Expert verified
The simplified form is \[ 4 \sqrt{3} \].

Step by step solution

01

Understand the Expression

Given the expression to simplify is \[ \sqrt[4]{48^2} \]. This means we need to find the fourth root of 48 squared.
02

Simplify the Power

First, simplify the power inside the root. Since \[ 48^2 = 48 \times 48 \], calculate that value. \[ 48 \times 48 = 2304 \]
03

Apply the Fourth Root

Now take the fourth root of the result obtained: \[ \sqrt[4]{2304} \]. To simplify this, look for factors or perfect squares within 2304. Rewriting 2304 as a product of its prime factors may help.
04

Prime Factorization

Perform prime factorization of 2304: \[ 2304 = 2^8 \times 3^2 \].
05

Simplify using the Fourth Root

Using the property of roots, \[ \sqrt[4]{2^8 \times 3^2} = \sqrt[4]{(2^2)^4 \times 3^2} = 2^2 \times \sqrt[4]{3^2} = 4 \times \sqrt[4]{9} \].
06

Simplify the Remaining Root

Recognize that \[ \sqrt[4]{9} = \sqrt[4]{3^2} = 3^{1/2} = \sqrt{3} \]. Therefore: \[ 4 \times \sqrt[4]{9} = 4 \times \sqrt{3} \]

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Key Concepts

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

Fourth Roots
When we talk about fourth roots, we are referring to a specific type of root. Just like the square root \( \sqrt\ \) finds a number that, when multiplied by itself twice, equals the original number, the fourth root \( \sqrt[4]{\} \) finds a number which, when multiplied by itself four times, results in the original value. To simplify a fourth root, look for values that can be evenly broken down into four equal parts.
Prime Factorization
Prime factorization is a method of breaking down a number into its prime components. To factorize a number, repeatedly divide it by the smallest possible prime number until all resulting factors are prime. For example, when handling the number 2304, we perform the following operations: \[ 2304 \div 2 = 1152 \ 1152 \div 2 = 576 \ 576 \div 2 = 288 \ 288 \div 2 = 144 \ 144 \div 2 = 72 \ 72 \div 2 = 36 \ 36 \div 2 = 18 \ 18 \div 2 = 9 \ 9 \div 3 = 3 \ 3 \div 3 = 1 \] This results in \[ 2304 = 2^8 \times 3^2 \] Once factorized, it's easier to manage the number under a root.
Radicals
A radical is a mathematical symbol representing the root of a number. The basic form is \( \sqrt{\} \), but it can be modified like \( \sqrt[n]{\} \) to indicate any root, such as the fourth root. Here's a brief overview of working with radicals:
  • ### Simplifying: Reduce the expression inside the radical by factoring.
  • ### Combining: When multiplying or dividing, you can combine under a single radical sign.
  • ### Breaking Down: For terms with exponents, separate and simplify. For instance, \sqrt[4]{2^8} = (2^2)^2= 4
Recognizing and simplifying radicals is key in algebra and higher math.

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Most popular questions from this chapter

Find the equation of a circle satisfying the given conditions. Center: (0,0)\(;\) radius: 9

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[3]{\sqrt[5]{\sqrt{y}}} $$

Graph each circle. Identify the center and the radius. \((x-2)^{2}+(y-3)^{2}=4\)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{6}{\sqrt{5}+\sqrt{3}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{5}+\sqrt{6}}{\sqrt{3}-\sqrt{2}} $$
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