**Title: Standard Form (Scientific Notation)**
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### What is Standard Form (Scientific Notation)?
Standard Form, or Scientific Notation, is a way of expressing very large or very small numbers conveniently. It is written in the form:
$$
a \times 10^n
$$
where:
– \( 1 \leq |a| < 10 \)
- \( n \) is an integer
This notation helps in simplifying operations with such numbers in various sciences and mathematical contexts.
---
##
Writing Numbers in Standard Form
To write a number in standard form, you express it as a product of a number between 1 and 10 and a power of ten.
### Easy Questions:
1. Convert 8000 into standard form.
**Solution**:
\(8000 = 8 \times 10^3\)
2. Convert 0.0025 into standard form.
**Solution**:
\(0.0025 = 2.5 \times 10^{-3}\)
### Medium Questions:
1. Write 62,500 in standard form.
**Solution**:
\(62,500 = 6.25 \times 10^4\)
2. Express 0.000457 as a number in scientific notation.
**Solution**:
\(0.000457 = 4.57 \times 10^{-4}\)
### Hard Questions:
1. Convert 98,000,000,000 into standard form.
**Solution**:
\(98,000,000,000 = 9.8 \times 10^{10}\)
2. Express 0.0000000135 in scientific notation.
**Solution**:
\(0.0000000135 = 1.35 \times 10^{-8}\)
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##
Converting from Standard Form
To convert from standard form to ordinary notation, multiply the number with the given power of ten.
### Easy Questions:
1. Convert \(3 \times 10^2\) into a normal number.
**Solution**:
\(3 \times 10^2 = 300\)
2. Convert \(7 \times 10^{-1}\) into ordinary form.
**Solution**:
\(7 \times 10^{-1} = 0.7\)
### Medium Questions:
1. Convert \(4.56 \times 10^3\) to an ordinary number.
**Solution**:
\(4.56 \times 10^3 = 4560\)
2. Write the value of \(6.73 \times 10^{-4}\) normally.
**Solution**:
\(6.73 \times 10^{-4} = 0.000673\)
### Hard Questions:
1. Convert \(9.91 \times 10^7\) to the standard decimal form.
**Solution**:
\(9.91 \times 10^7 = 99,100,000\)
2. Express \(3.005 \times 10^{-6}\) as a regular number.
**Solution**:
\(3.005 \times 10^{-6} = 0.000003005\)
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##
Operations with Numbers in Standard Form
You can multiply, divide, add, and subtract numbers in scientific notation by following exponent rules and aligning powers of ten.
### Easy Questions:
1. Multiply \(2 \times 10^3\) and \(3 \times 10^2\).
**Solution**:
\((2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^5\)
2. Divide \(5 \times 10^6\) by \(1 \times 10^2\).
**Solution**:
\((5 \times 10^6) / (1 \times 10^2) = 5 \times 10^4\)
### Medium Questions:
1. Add \(3.5 \times 10^4\) and \(4.2 \times 10^4\).
**Solution**:
\( (3.5 + 4.2) \times 10^4 = 7.7 \times 10^4\)
2. Subtract \(8.1 \times 10^5\) from \(1.4 \times 10^6\).
**Solution**:
\(1.4 \times 10^6 – 8.1 \times 10^5 = 5.9 \times 10^5\)
### Hard Questions:
1. Multiply \(6.2 \times 10^{-3}\) with \(5.1 \times 10^4\).
**Solution**:
\(6.2 \times 5.1 = 31.62;\)
Adjust to standard form:
\(3.162 \times 10^1\)
So,
\(3.162 \times 10^1 \times 10^{-3 + 4} = 3.162 \times 10^2\)
2. Divide \(7.5 \times 10^5\) by \(2.5 \times 10^2\).
**Solution**:
\((7.5 / 2.5) = 3;\)
\(10^{5 – 2} = 10^3\)
So, answer = \(3 \times 10^3\)
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##
Standard Form in Real-world Applications
Scientists, engineers, and financial analysts use scientific notation to convey large or small values, like distances, particle sizes, or transaction values.
### Easy Questions:
1. The diameter of an atom is about \(5 \times 10^{-10}\) meters. Express in decimal form.
**Solution**:
\(5 \times 10^{-10} = 0.0000000005\)
2. A population of bacteria grows to \(2 \times 10^7\). Express in decimal.
**Solution**:
\(2 \times 10^7 = 20,000,000\)
### Medium Questions:
1. The sun is approximately \(1.496 \times 10^8\) km from Earth. Write in decimal.
**Solution**:
\(1.496 \times 10^8 = 149,600,000\)
2. One molecule of water weighs around \(2.99 \times 10^{-23}\) grams.
**Solution**:
\(2.99 \times 10^{-23} = 0.0000000000000000000000299\)
### Hard Questions:
1. A light-year is roughly \(9.461 \times 10^{12}\) km. Multiply by 4 (light years).
**Solution**:
\(4 \times 9.461 = 37.844\);
So, \(3.7844 \times 10^1 \times 10^{12} = 3.7844 \times 10^{13}\)
2. The mass of a virus is about \(3.2 \times 10^{-19}\) kg. Multiply by 150 virus particles.
**Solution**:
\(3.2 \times 150 = 480\);
\(4.8 \times 10^2 \times 10^{-19} = 4.8 \times 10^{-17}\)
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### Why is it Important to Study Standard Form (Scientific Notation)?
Understanding how to work with scientific notation helps students solve real-world problems in science and engineering. It makes it easier to handle large-scale and microscopic values with precision and speed.
Notable recent mathematicians and scientists working in fields requiring such notation include:
– Maryna Viazovska (Fields Medalist, works with high-dimensional spaces)
– Terence Tao (Known for number theory and mathematical analysis)
– Kip Thorne (Theoretical physicist using scientific notation in space-time analysis)
Learning and applying scientific notation opens doors in STEM fields—astronomy, physics, computer science, and finance.
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