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Chapter 10: Problem 68

Rewrite in decimal form. $$ 8.17 \times 10^{7} $$

Short Answer

Expert verified
The decimal form of \(8.17 \times 10^{7}\) is \(81700000.0\).

Step by step solution

01

Identify Base and Exponent

The base number is \(8.17\) and the exponent is \(7\). An exponent of \(7\) in the context of scientific notation means we will move the decimal point \(7\) places to the right.
02

Move the Decimal Point

Starting from the given base number \(8.17\), move the decimal point \(7\) places to the right. For each place beyond the given precision of \(8.17\), add a zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scientific Notation
Scientific notation is a method of writing very large or very small numbers in a compact form. It's fundamentally composed of two parts: a coefficient and an exponent. The coefficient is typically a number greater than or equal to 1 and less than 10, and it's accompanied by an exponent, which tells us how many places to move the decimal point to obtain the value in standard form. For example, in the notation
\(8.17 \times 10^{7}\),
\(8.17\) acts as the coefficient and the number 7 is the exponent of the base 10. This method is widely used in mathematics, science, and engineering to efficiently communicate and manipulate extremely large or small numerical values.
Decimal Conversion
Converting from scientific notation to decimal form entails adjusting the position of the decimal point relative to the value of the exponent. When converting a number such as
\(8.17 \times 10^{7}\),
you need to shift the decimal point to the right since the exponent is positive. If the exponent were negative, the decimal point would instead move to the left. It's crucial to note that this process sometimes introduces zeroes to fill in places, ensuring that the decimal point moves the correct number of places. This conversion is essential for understanding and using numbers in everyday contexts where standard decimal notation is required.
Base and Exponent
The concept of base and exponent is rooted in the operation of exponentiation, where a number (the base) is multiplied by itself several times as indicated by the exponent. In our example, the base is
\(8.17\),
and the exponent is 7, meaning that the base would be used as a factor 7 times in multiplication. However, in scientific notation, rather than actually multiplying the base several times, we use the exponent to indicate how many positions to move the decimal point. If the exponent is positive, we move to the right; if it's negative, the movement is toward the left.
Place Value
Place value is a fundamental concept in mathematics that refers to the value of each digit in a number, depending on its position. As we shift the decimal point in scientific notation conversion, we are effectively changing the place value of digits in the number. For instance, in the expression
\(8.17 \times 10^{7}\),
moving the decimal 7 places to the right transforms \(8.17\) into 81,700,000. Each zero added occupies a new place value, signifying lesser values as you move from left to right. Understanding place value is critical for correctly performing decimal conversions and for grasping the significance of number magnitude in various contexts.

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