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Magazine Chapter 1: Problem 14
Combined Operations with Exact Numbers. Perform each computation by calculator. $$\sqrt{434+466}$$
Short Answer
Expert verified
The result of \(\sqrt{434+466}\) is 30.
Step by step solution
01
Add the numbers under the square root
Start by computing the sum of the numbers inside the square root: add 434 and 466 together.
02
Calculate the sum
Adding the numbers yields 434 + 466 = 900.
03
Apply the square root
Now, take the square root of the result. The square root of 900 is a number which, when multiplied by itself, gives 900.
04
Find the square root
Since 30 multiplied by 30 equals 900, the square root of 900 is 30.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combined Operations
Understanding combined operations in mathematics is fundamental to problem-solving across various fields. It involves performing two or more arithmetic operations, such as addition, subtraction, multiplication, or division, in a sequence to solve an equation or expression. In our textbook exercise, we encountered a problem that required us to add numbers first and then compute a square root, showing a simple instance of combined operations.
Here's how to tackle combined operations effectively:
Here's how to tackle combined operations effectively:
- Identify the operations involved and the sequence they must be performed in.
- Carefully follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
- Perform each operation step by step, ensuring accuracy at each stage to prevent errors that can affect the final result.
Square Root Computation
Square root computation is a concept that involves finding a number, which when multiplied by itself, gives the original number. The square root of a number 'x' is often represented as \(\sqrt{x}\).
In the context of our exercise \(\sqrt{434+466}\), we first simplified the expression under the radical by adding both numbers to get 900. The next step was to find the square root of 900, which is 30 since \(30\times30=900\).
Here are some practical tips for square root computation:
In the context of our exercise \(\sqrt{434+466}\), we first simplified the expression under the radical by adding both numbers to get 900. The next step was to find the square root of 900, which is 30 since \(30\times30=900\).
Here are some practical tips for square root computation:
- Memorize the square roots of perfect squares (e.g., \(\sqrt{1}=1\), \(\sqrt{4}=2\), \(\sqrt{9}=3\), etc.) for quick reference.
- When dealing with non-perfect squares, estimate the square root by finding the nearest perfect squares.
- Use the long division method for square roots or a calculator for a more accurate and faster result.
Calculator Usage for Mathematics
A calculator is an indispensable tool for students as it aids in accurate and efficient computation, especially for complex and time-consuming calculations. While working through the exercise of combined operations involving square roots, using a calculator allows us to quickly verify results and understand the steps of our calculations.
To effectively use a calculator for mathematics, keep these points in mind:
To effectively use a calculator for mathematics, keep these points in mind:
- Ensure the calculator is in the correct mode for the calculations you are performing (e.g., degree or radian mode for trigonometry).
- Practice with the calculator to become familiar with its functions, such as memory recall or scientific notations.
- Understand the limitations of your calculator; not all are capable of symbolic algebra or graphing functions.
Mathematical Problem Solving
Mathematical problem solving is more than just crunching numbers; it is about understanding patterns, conceptualizing strategies, and logically working through steps to achieve a solution. The exercise we've discussed illustrates the problem-solving process in a straightforward manner鈥攖he first step is to comprehend the problem, followed by strategizing how to approach and execute the solution.
Here are some key points in mathematical problem solving:
Here are some key points in mathematical problem solving:
- Read and understand the problem thoroughly before starting.
- Identify all the information given and determine what the problem is asking for.
- Decide on the order of operations and tools needed, such as a calculator.
- Work through the problem step by step and check your work to avoid small mistakes.
