Express in scientific notation:
[tex]\frac{\left(7 \times 10^{-4}\right)}{\left(3 \times 10^4\right)\left(5 \times 10^5\right)}=[/tex]
Answer (1)
Simplify the expression by separating numerical and exponential parts: 15 7 × 1 0 4 × 1 0 5 1 0 − 4 .
Calculate the numerical part: 15 7 ≈ 0.4667 .
Calculate the exponential part: 1 0 9 1 0 − 4 = 1 0 − 13 .
Combine and convert to scientific notation: 0.4667 × 1 0 − 13 = 4.667 × 1 0 − 14 ≈ 5 × 1 0 − 14 . However, this is not in the options. There might be a typo in the options.
Explanation
Understanding the Problem We are asked to express the result of the expression ( 3 × 1 0 4 ) ( 5 × 1 0 5 ) ( 7 × 1 0 − 4 ) in scientific notation and choose the correct answer from the given options.
Simplifying the Expression First, let's simplify the expression by separating the numerical and exponential parts: ( 3 × 1 0 4 ) ( 5 × 1 0 5 ) ( 7 × 1 0 − 4 ) = 3 × 5 7 × 1 0 4 × 1 0 5 1 0 − 4 Now, we simplify the numerical part: 3 × 5 7 = 15 7 And simplify the exponential part: 1 0 4 × 1 0 5 1 0 − 4 = 1 0 4 + 5 1 0 − 4 = 1 0 9 1 0 − 4 = 1 0 − 4 − 9 = 1 0 − 13
Converting to Scientific Notation Now, combine the numerical and exponential parts: 15 7 × 1 0 − 13 We need to convert the numerical part to scientific notation. We calculate 15 7 : 15 7 ≈ 0.4666... To express this in scientific notation, we write it as: 0.4666... = 4.666... × 1 0 − 1
Adjusting the Exponent and Rounding Now, we adjust the exponent: 4.666... × 1 0 − 1 × 1 0 − 13 = 4.666... × 1 0 − 1 − 13 = 4.666... × 1 0 − 14 We need to round the numerical part to one significant figure, as the options are given in that format. Since 4.666... is closer to 5 than to 4 , we round it to 5 . Therefore, the expression in scientific notation is: 5 × 1 0 − 14
Final Answer and Conclusion However, none of the given options match our result. Let's re-examine the options and our calculations. The given options are: 0.47 × 1 0 − 17 5 × 1 0 − 48 0.5 × 1 0 − 17
It seems there was an error in the previous steps. Let's recalculate the exponent part: 1 0 4 × 1 0 5 1 0 − 4 = 1 0 − 4 − 4 − 5 = 1 0 − 13 Then, we have 15 7 × 1 0 − 13 ≈ 0.4667 × 1 0 − 13 . To write this in scientific notation, we have 4.667 × 1 0 − 1 × 1 0 − 13 = 4.667 × 1 0 − 14 . Rounding to one significant figure gives 5 × 1 0 − 14 . Still, this is not among the options.
Let's check the initial expression again. It is ( 3 × 1 0 4 ) ( 5 × 1 0 5 ) 7 × 1 0 − 4 . So, 15 × 1 0 9 7 × 1 0 − 4 = 15 7 × 1 0 − 4 − 9 = 15 7 × 1 0 − 13 ≈ 0.4667 × 1 0 − 13 . In scientific notation, this is 4.667 × 1 0 − 14 , which rounds to 5 × 1 0 − 14 .
It seems there's a mistake in the options provided. The correct answer should be approximately 5 × 1 0 − 14 . However, if we must choose from the given options, the closest one in terms of magnitude is 0.5 × 1 0 − 17 , but it's significantly different. There might be a typo in the question or the options.
Examples
Scientific notation is used in many fields, such as physics, astronomy, and engineering, to represent very large or very small numbers. For example, the distance to the nearest star, Proxima Centauri, is approximately 4.0175 × 1 0 16 meters. Similarly, the mass of an electron is approximately 9.109 × 1 0 − 31 kilograms. Using scientific notation makes it easier to work with these numbers and perform calculations.
