Index Laws and Scientific Notation

What you’ll learn

Combine powers and express large or small numbers clearly.

ប្រើច្បាប់ស្វ័យគុណ និងសរសេរចំនួនជាទម្រង់វិទ្យាសាស្ត្រ។

Meaning and use

An exponent counts repeated multiplication for positive integer powers. Product and quotient rules require the same base. Scientific notation writes a nonzero number as a coefficient times a power of ten, with the coefficient’s absolute value from one up to but not including ten.

ស្វ័យគុណវិជ្ជមានបង្ហាញការគុណចំនួនដដែលៗ។ ច្បាប់បូកឬដកនិទស្សន្តត្រូវមានគោលដូចគ្នា។ ទម្រង់វិទ្យាសាស្ត្រមានមេគុណដែលតម្លៃដាច់ខាតពី 1 ដល់តិចជាង 10។

Formulas

Same-base laws

\[a^m a^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n}\]

Zero, negative powers, and notation

\[a^0=1,\quad a^{-n}=\frac1{a^n},\quad N=c\times10^k\]

Symbols and units

Symbol	Meaning	Unit or restriction
a	Base	Nonzero for quotient, zero-power, and negative-power rules
m, n, k	Integer exponents	Dimensionless
c	Scientific-notation coefficient	1 ≤ abs(c) < 10 for nonzero N

Conditions and limits

• Do not combine powers with different bases by adding exponents.
• Zero cannot be raised to a negative power; 0⁰ is not assigned a value here.
• និទស្សន្តអវិជ្ជមានមានន័យចំនួនច្រាស មិនមែនចម្លើយអវិជ្ជមានទេ។

Worked examples

Same-base product

Simplify \(2^3\times2^4\).

1. Add exponents: \(2^{3+4}=2^7\).
2. Evaluate \(2^7=128\).

128

គោលដូចគ្នា ទើបអាចបូកនិទស្សន្តពេលគុណ។

A small number

Write 0.0045 in scientific notation.

1. Move the decimal three places right to make 4.5.
2. Compensate using \(10^{-3}\).

\(4.5\times10^{-3}\)

ចំនួនតូចជាង 1 ប្រើនិទស្សន្តអវិជ្ជមានក្នុងករណីនេះ។

Common mistakes

Instead of	Use	Why
2³ × 2⁴ = 2¹²	2³ × 2⁴ = 2⁷	Add exponents when multiplying same-base powers.
10⁻² = −100	10⁻² = 1/100	A negative exponent indicates a reciprocal.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Simplify 5² × 5³.
2. Evaluate 3⁰.
3. Write 62000 in scientific notation.

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 5⁵ = 3125

   Same base: add 2 + 3.

2. 1

   The base is nonzero.

3. 6.2 × 10⁴

   The coefficient is between 1 and 10.

Quick recap

• Check that bases match.
• Negative powers are reciprocals.
• Normalise the scientific-notation coefficient.