Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 73
Find a rational approximation, to the nearest tenth, for each radical expression. (Objective 2) $$9 \sqrt{5}-3 \sqrt{5}$$
Short Answer
Expert verified
13.4
Step by step solution
01
Understand the Expression
The given expression is \(9 \sqrt{5} - 3 \sqrt{5}\). Both terms contain \(\sqrt{5}\), so they can be combined by subtracting the coefficients.
02
Subtract the Coefficients
Subtract the coefficients of the square root terms: \(9 - 3 = 6\). Therefore, \(9 \sqrt{5} - 3 \sqrt{5} = 6 \sqrt{5}\).
03
Estimate \(\sqrt{5}\)
Find an estimate for \(\sqrt{5}\). It is approximately \(2.236\) since \(2.2^2 = 4.84\) (less than 5) and \(2.3^2 = 5.29\) (more than 5).
04
Multiply by the Coefficient
Now multiply the coefficient 6 by \(\sqrt{5}\) to get \(6 \times 2.236 = 13.416\).
05
Round to the Nearest Tenth
Round \(13.416\) to the nearest tenth, which gives us \(13.4\).
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.
Rational Approximation
Rational approximation is a technique used to find a number that is close to a given irrational number using rational numbers (numbers that can be expressed as a fraction with an integer numerator and a non-zero integer denominator). This is particularly useful when dealing with square roots and other mathematical computations that involve irrational numbers.
Here鈥檚 how it can be done:
Here鈥檚 how it can be done:
- Identify the irrational number you wish to approximate, such as a square root. In this case, it is \( \sqrt{5} \).
- Estimate its value to a manageable decimal form (e.g., 2.236).
- This decimal approximation can be treated as a rational approximation for practical purposes.
Radical Expressions
A radical expression is an expression that includes a root, such as a square root, cube root, etc. They are commonly used in algebra to solve equations and model real-world scenarios.
When working with radical expressions, it's important to understand the following:
When working with radical expressions, it's important to understand the following:
- The radicand is the number or expression inside the radical symbol (鈭�).
- The index denotes the degree of the root (e.g., \( \sqrt[n]{x} \) represents the n-th root of x, with square roots having an implied index of 2).
- Radical expressions can often be simplified by combining like terms, such as terms involving \( \sqrt{5} \) in our original problem.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. Finding square roots is a common operation in mathematics and is essential in various areas, from solving quadratic equations to calculating geometric properties.
Useful details about square roots:
Useful details about square roots:
- A positive number's principal square root is always non-negative; \( \sqrt{25} \) equals 5, not -5.
- Square roots of non-perfect squares are irrational numbers, meaning they cannot be expressed as exact fractions (e.g., \( \sqrt{5} \)).
- Approximating square roots, as done in this problem where \( \sqrt{5} \) approximates to 2.236, is a practical step in many problems.
Mathematical Estimation
Mathematical estimation is a useful tool that involves finding an approximate value of a mathematical expression or number, often to simplify calculations or gain insights without a need for precise values.
Key points about estimation:
Key points about estimation:
- Estimation helps work with complex numbers more easily by using rounded or simplified numbers for quicker calculations.
- It's common to estimate when dealing with irrational numbers, such as in this example with \( \sqrt{5} \).
- Estimation is useful in determining the "closeness" to the actual number, which assists in practicality for real-world applications.
