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How to Convert 0.5 × 10⁷ to Standard Scientific Notation
Mathematics
Grade 9 (Junior High School)
Question Content
Convert 0.5 × 10⁷ into standard form.
Correct Answer
5 × 10⁶
Detailed Solution Steps
1
Step 1: Recall the definition of standard form (scientific notation): a number is in standard form when it is written as \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer.
2
Step 2: Adjust the coefficient 0.5 to meet the requirement \(1 \leq a < 10\). Multiply 0.5 by 10 to get 5, which falls in the valid range for \(a\).
3
Step 3: To keep the value of the number unchanged, we need to divide the power of 10 by 10. Dividing \(10^7\) by 10 is equivalent to subtracting 1 from the exponent, so \(10^7\) becomes \(10^{7-1} = 10^6\).
4
Step 4: Combine the adjusted coefficient and the adjusted power of 10 to get the standard form: \(5 \times 10^6\).
Knowledge Points Involved
1
Standard Form (Scientific Notation) Definition
Standard form (or scientific notation) is a way to write very large or very small numbers compactly, expressed as \(a \times 10^n\). Here, \(a\) must be a real number where \(1 \leq |a| < 10\), and \(n\) is an integer. This format makes it easier to compare, calculate, and communicate large/small values in science, engineering, and mathematics.
2
Adjusting Coefficients in Scientific Notation
When the coefficient \(a\) is outside the range \(1 \leq |a| < 10\), you must adjust it by multiplying or dividing by powers of 10. If you multiply the coefficient by 10, you subtract 1 from the exponent of 10; if you divide the coefficient by 10, you add 1 to the exponent of 10. This ensures the overall value of the number remains unchanged.
3
Rules for Manipulating Exponents of 10
When dividing powers of 10 with the same base, subtract the exponents: \(10^m \div 10^k = 10^{m-k}\). This rule is used to adjust the exponent when modifying the coefficient in scientific notation, maintaining the equivalence of the original and rewritten number.
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