Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 56
Simplify. See Examples 3 and 4 $$ \sqrt{12 r^{9} s^{12}} $$
Short Answer
Expert verified
The expression simplifies to \( 2r^4s^6\sqrt{3r} \).
Step by step solution
01
Prime Factorization of 12
To simplify the expression, we start by simplifying the square root of 12. First, we express 12 as a product of its prime factors: \( 12 = 2^2 \times 3 \). This will help us find the square root.
02
Split the Radicand
Next, split the original expression inside the square root as separate components: \( \sqrt{12 r^9 s^{12}} = \sqrt{2^2 \times 3 \times r^9 \times s^{12}} \). Now, consider each factor separately.
03
Simplify Square Root of Numbers
Take the square root of the numerical part inside the square root: \( \sqrt{2^2 \times 3} = 2\sqrt{3} \), because the square root of \( 2^2 \) is 2, and 3 stays under the square root.
04
Simplify Variables with Even Powers
Apply the square root to \( s^{12} \): Since the exponent is even, \( \sqrt{s^{12}} = s^6 \). This is because \( s^{12} = (s^6)^2 \), and the square root of \( (s^6)^2 \) is \( s^6 \).
05
Simplify Variables with Odd Powers
Deal with \( r^9 \): Since 9 is odd, express \( r^9 = r^8 \times r \) where \( r^8 = (r^4)^2 \). The square root of \( r^8 \) is \( r^4 \), thus, \( \sqrt{r^9} = r^4 \sqrt{r} \).
06
Combine All Simplified Components
Combine all parts together: \( \sqrt{12 r^9 s^{12}} = 2 r^4 s^6 \sqrt{3r} \). Therefore, the expression is simplified successfully.
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 the process of breaking down a composite number into its prime number components. A prime number is a number greater than one that has no divisors besides one and itself.
To simplify a square root, like the square root of 12, we first find the prime factorization of 12.
To simplify a square root, like the square root of 12, we first find the prime factorization of 12.
- Start by dividing 12 by the smallest prime number, which is 2. We get 12 = 2 脳 6.
- Keep dividing by 2 until you can no longer divide evenly. So, 6 can be divided by 2 to get 3. Now, 12 is expressed as 2 脳 2 脳 3.
- This translates to 12 = 2虏 脳 3 in terms of prime factors.
Square Root Properties
Understanding the properties of square roots is essential when simplifying expressions. A square root surrounds a number or a variable, and the goal is to simplify it by extracting perfect squares.
Here are some basic properties:
Here are some basic properties:
- The square root of a product can be split into the product of square roots: \( \sqrt{ab} = \sqrt{a}\sqrt{b} \).
- The square root of a square is the original number: \( \sqrt{a^2} = a\).
- If a factor inside a square root is a perfect square, you can take it out of the square root.
Simplifying Variable Expressions
When simplifying expressions that include variables under a square root, it's important to consider the exponents of the variables.
Here's how you simplify based on the exponent's parity:
Here's how you simplify based on the exponent's parity:
- If the exponent is even, you can fully take that variable out of the square root. For example, \( \sqrt{s^{12}} = s^6 \) since \( s^{12} = (s^6)^2 \).
- If the exponent is odd, separate it into an even part and a leftover 1. For example, \( r^9 = r^8 \times r = (r^4)^2 \times r \), and hence \( \sqrt{r^9} = r^4\sqrt{r} \).
Mathematical Expressions
A mathematical expression consists of numbers, variables, and different operations. Understanding how to simplify such expressions is key for solving problems efficiently.
Mathematical expressions often involve:
Mathematical expressions often involve:
- Numbers, as coefficients or standalone integers.
- Variables, which stand for unspecified numbers.
- Operations like addition, subtraction, multiplication, or division.
