Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 32
Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{28 x^{4} y^{12}}\)
Short Answer
Expert verified
The simplest radical form is \( 2x^2y^6\sqrt{7} \).
Step by step solution
01
Prime Factorization of the Coefficient
First, we factor the coefficient 28 under the square root into its prime factors. \( 28 = 2^2 \times 7 \). Knowing this helps us to simplify the radical expression later on.
02
Separate the Radicand
We can write the expression under the square root as the multiplication of separate parts: \( \sqrt{(2^2 \times 7) \times x^4 \times y^{12}} \). This separates the numerical part and the variable parts.
03
Simplify Each Part Under the Square Root
Simplify the square root of each part: \( \sqrt{2^2} = 2 \), \( \sqrt{x^4} = x^2 \), and \( \sqrt{y^{12}} = y^6 \).Combine these simplified parts: \( 2x^2y^6 \). The \( \sqrt{7} \) part remains under the radical because it cannot be simplified further.
04
Write the Simplified Expression
Combine all simplified parts to form the final expression in simplest radical form: \( 2x^2y^6\sqrt{7} \). This expression is the simplified form of the original radical expression.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Prime Factorization
Prime factorization is a method where we break down a number into its smallest building blocks: prime numbers. A prime number is a whole number greater than 1 whose only factors are 1 and itself. For example, the number 28 can be expressed as the product of prime numbers by dividing it repeatedly until only prime numbers remain.
- The prime factorization of 28 is achieved by first dividing it by the smallest prime, 2, to get 14.
- Divide 14 by 2 again, which gives 7, a prime number.
- The process ends there, as 7 cannot be divided further by any number other than 1 and itself.
Radical Expressions
Radical expressions are mathematical expressions that contain a radical symbol, \(\sqrt{}\), which is used to denote roots, such as square roots or cube roots. They're used to represent numbers that can be expressed as a fraction or decimal when squared or otherwise raised. For instance, the square root of 9, written as \(\sqrt{9}\), equals 3.
- Radicals can be applied to variables as well, as seen with expressions like \(\sqrt{x^4}\) or \(\sqrt{y^{12}}\).
- Simplifying radical expressions often involves performing operations, such as multiplication or division, within the radical.
- The aim is to reduce the expression to its simplest form, minimizing any numbers inside the radical.
Simplifying Radicals
Simplifying radicals involves the process of rewriting a radical expression in its simplest form. This means reducing the number inside the radical symbol as much as possible, while also simplifying any coefficients and variable parts.
- Simplifying involves separating the expression into its factorized parts using prime factorization and then simplifying each individual part.
- For instance, the term \(\sqrt{28x^4y^{12}}\) can be viewed as several separate elements: \(2^2\), \(x^4\), and \(y^{12}\).
- We take the square root of each of these parts: \(\sqrt{2^2} = 2\), \(\sqrt{x^4} = x^2\), and \(\sqrt{y^{12}} = y^6\).
Real Numbers
Real numbers are all the numbers on the number line, including both rational numbers (like \(1/2\), 3, and 4.75) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)). They include whole numbers, integers, fractions, and decimals, basically covering all numbers we can think of in common arithmetic.
- In the context of radical expressions, variables are considered to represent only positive real numbers. This ensures that we are dealing with valid real values under the square root.
- For example, in \(\sqrt{28x^4y^{12}}\), if \(x\) and \(y\) are positive real numbers, no negative results will appear from the square roots of even powers.
- Real numbers provide a broad framework to understand the arithmetic behind simplifying radicals and evaluating different expressions involving square roots.
