Example Question - scientific notation
Here are examples of questions we've helped users solve.
Converting Large Number to Scientific and Floating-Point Notation
<p>To write the number "googol" in scientific notation, we recognize that a googol is defined as \(10^{100}\). Thus in scientific notation, it is simply:</p> <p>\(1 \times 10^{100}\)</p> <p>For the floating-point notation, which is typically used in calculators and computers for representing numbers, it would be impractical to display all the digits of a googol. However, depending on the calculator's display, it could be approximated to a certain number of significant digits. Since the options given are limited and none can fully represent a googol, there's no perfect match, but the understanding is that we're looking for the representation closest to \(10^{100}\).</p> <p>The closest approximation from the options for a floating-point notation is \(1 \times 10^{100}\), which aligns with the scientific notation.</p>
Expressing a Large Number in Scientific Notation
<p>Write \( 450,000,000 \) in scientific notation:</p> <p>\( 450,000,000 = 4.5 \times 10^8 \)</p> <p>Therefore, the floating-point form is:</p> <p>\( 4.5 \) and the exponent is \( 08 \).</p>
Identifying the Equivalent Floating-Point Representation
<p>The number given in the image is \(2 \times 10^{-5}\).</p> <p>To express this number in the decimal form, we shift the decimal point 5 places to the left because the exponent is -5.</p> <p>Therefore, the equivalent floating-point representation is:</p> <p>\(0.00002\), which can be rewritten as \(0.2 \times 10^{-4}\).</p> <p>The correct answer is \(0.2 \times 10^{-4}\).</p>
Converting Between Scientific Notation and Floating-Point Notation
<p>The mathematical concept requested in the image involves understanding and converting between scientific notation and floating-point notation. Typically, any number in scientific notation is written as \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer.</p> <p>To convert from floating-point to scientific notation, we must adjust the decimal point to fit the criteria for \( a \), and then determine the appropriate exponent \( n \) that \( 10 \) must be raised to.</p> <p>Looking at the options provided:</p> <p>\( 20 \times 10^{-06} \) can be adjusted to \( 2.0 \times 10^{1} \times 10^{-06} = 2.0 \times 10^{-05} \)</p> <p>\( 0.02 \times 10^{-03} \) can be adjusted to \( 2.0 \times 10^{-02} \times 10^{-03} = 2.0 \times 10^{-05} \)</p> <p>\( 2 \times 10^{-05} \) is already in scientific notation.</p> <p>Without any additional context or numerical values in the image to match the options against, all three options provided are valid representations of numbers in scientific notation adjusted from the given floating-point notation starting points.</p>
Converting a Number to Scientific Notation and Floating-Point Notation
<p>For scientific notation, move the decimal point in 21.79 until there is one non-zero digit to the left of the decimal point:</p> <p>\(2.179 \times 10^1\)</p> <p>For floating-point notation, this is the same as the original number:</p> <p>\(21.79\)</p>
Mathematical Expression Rounding in Scientific Notation
The image shows a mathematical expression where a number is rounded in scientific notation. The original number is 10470000, and it is being rounded to one significant figure, becoming \(1 \times 10^7\). The steps shown are correct. The expression starts with the number 10470000 and then brackets are placed to separate the significant figure (1) from the rest of the number (0470000). Next, the expression is rewritten with the rounded number as \(1 \times 10^7\). The chinese character "万" translates to "ten thousand" in English, which is part of the Chinese numbering system where 10,000 is a basic unit.
Converting a Number to Scientific Notation
The number 725,000,000 can be written in scientific notation by placing the decimal point after the first non-zero digit and then counting how many places the decimal has moved. This number is written as \( 7.25 \times 10^n \), where \( n \) is the number of places the decimal moved. For the number 725,000,000, the decimal point moves 8 places to the left to get from 725,000,000.0 to 7.25. Therefore, the number 725,000,000 would be written in scientific notation as: \[ 7.25 \times 10^8 \]
Converting Decimal to Scientific Notation
To express the number 0.0736 in scientific notation, you need to rewrite it as a product of a number between 1 and 10 and a power of 10. Move the decimal point so that there is one non-zero digit to the left of the decimal. For the number 0.0736, you move the decimal two places to the right to get 7.36. This is the number between 1 and 10. Now, record the number of places you moved the decimal point as a negative exponent of 10 (because the original number is less than 1). Since you moved the decimal two places, the exponent will be -2. So, the number 0.0736 in scientific notation is: \[7.36 \times 10^{-2}\]
Arithmetic Problem Solution in Scientific Notation
The question in the image is an arithmetic problem involving multiplication of numbers in scientific notation: 7 x 10^9 To solve this, you simply multiply the number 7 by 10 raised to the power of 9. Since 10^9 represents the number 1 followed by 9 zeros, the multiplication is straightforward: 7 x 1,000,000,000 = 7,000,000,000 So the answer is 7 billion.
Determining Closest Planets by Distance
According to the table provided, we need to compare the distances between different pairs of planets to determine which pair is closest to each other. To do this, we look at the 'Distance (km)' column and find the smallest number. Here are the distances provided for each pair of planets in scientific notation: - Mercury to Saturn: 1.37 x 10^9 km - Venus to Jupiter: 6.72 x 10^8 km - Earth to Neptune: 4.35 x 10^9 km - Saturn to Jupiter: 3.46 x 10^8 km - Earth to Mars: 7.83 x 10^7 km When comparing numbers in scientific notation, you can compare the exponents first. The smallest exponent here is 7 (in 7.83 x 10^7), which corresponds to the Earth to Mars distance. The smaller the exponent, the smaller the number. Therefore, the smallest distance among the given options is the one between Earth and Mars, which is 7.83 x 10^7 km. This means the correct answer is Earth and Mars.
Understanding Scientific Notation for Hair Strands
The question seems to be incomplete, but it starts with the statement, "There are about 1 x 10^5 strands of hair on the human head." This statement is giving a scientific notation for the average number of hair strands on a human head, which is expressed as 1 times 10 to the power of 5. To convert this scientific notation to a standard number, you calculate 10^5 which is 10 multiplied by itself 4 more times: 10 x 10 x 10 x 10 x 10 = 100,000 Thus, the statement expresses that there are approximately 100,000 strands of hair on the average human head. Without the complete question, I cannot provide further assistance, but this explanation should help you understand the information presented in the image. If there was a specific question related to this statement, please provide it so I can help you further.
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