Chapter 1: Problem 62
Evaluate each expression. Retain the proper number of significant digits in your answer. Roots Find each principal root without using your calculator. $$\sqrt{25}$$
Short Answer
Expert verified
\sqrt{25} = 5
Step by step solution
01
Identify the type of root
The symbol \( \sqrt{ } \) represents the principal or positive square root of a number. To evaluate \( \sqrt{25} \) without a calculator, identify it as asking for the positive square root of 25.
02
Find the square root
Recall that the square root of a number is a value that, when multiplied by itself, gives the original number. The number 25 is a perfect square, as it can be expressed as \( 5 \times 5 \) or \( (-5) \times (-5) \) since \( 5^2 = 25 \) and \( (-5)^2 = 25 \) as well. However, since we're looking for the principal root, we choose the positive value.
03
Write the solution
The principal square root of 25 is 5. The answer should retain the proper number of significant digits. Since 25 has two significant digits, the answer should as well, which in this case does not change the result.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Significant Digits
Understanding significant digits, also known as significant figures, is vital in scientific notation to ensure precision and accuracy of calculations. These are the digits in a number that are necessary to express its precision. For example, in the number 25, there are two significant digits, '2' and '5'. If you have a number like 300, it may have one, two, or three significant digits, depending on whether the zeros are considered significant, which can depend on whether a decimal is present or if it is measured data.
In the context of square root calculations, it's essential to maintain the correct number of significant digits in your answer to reflect the number's precision. If the original number has two significant digits, like 25, the square root should also have two significant digits if possible. However, in the case of perfect squares, like 25, the square root will be an integer, and the concept of significant digits doesn't further change the result.
In the context of square root calculations, it's essential to maintain the correct number of significant digits in your answer to reflect the number's precision. If the original number has two significant digits, like 25, the square root should also have two significant digits if possible. However, in the case of perfect squares, like 25, the square root will be an integer, and the concept of significant digits doesn't further change the result.
Square Root Calculation
A square root calculation involves finding a number which, when multiplied by itself, results in the original number. When performing such calculations, especially with perfect squares, identifying whether the square root is a whole number or a decimal is crucial.
For non-perfect squares, estimating the square root may involve calculating the squares of numbers that are close to the square root and refining the estimate. There are methods such as prime factorization or using the long division method to accurately compute square roots. These techniques allow you to approximate square roots to several decimal places without needing a calculator.
For non-perfect squares, estimating the square root may involve calculating the squares of numbers that are close to the square root and refining the estimate. There are methods such as prime factorization or using the long division method to accurately compute square roots. These techniques allow you to approximate square roots to several decimal places without needing a calculator.
Perfect Squares
Perfect squares are integers that are the square of an integer. For example, 1, 4, 9, 16, and 25 are all perfect squares. Recognizing perfect squares quickly allows for rapid and accurate calculation of their square roots. This is particularly useful when performing non-calculator computation, as it eliminates the need for complex algorithms or approximations.
In educational settings, recognizing perfect squares leads to greater ease when simplifying radical expressions, factoring quadratic equations, and solving problems involving area and volume. The principal square root of a perfect square is always a positive integer, making them more straightforward to handle.
In educational settings, recognizing perfect squares leads to greater ease when simplifying radical expressions, factoring quadratic equations, and solving problems involving area and volume. The principal square root of a perfect square is always a positive integer, making them more straightforward to handle.
Non-calculator Computation
Non-calculator computation refers to calculating mathematical problems without the assistance of a calculator. This skill boosts mental arithmetic and promotes a deeper understanding of numeric relationships and mathematical principles. To compute square roots without a calculator, you can use methods such as:
- Estimation and repeated testing
- Prime factorization to find square factors
- The Babylonian method, also known as Heron's method