Power Play · Lesson 4 of 7
Powers of 10
“Use powers of 10 for place value, scientific notation and large-number comparison.”
• Express place value using powers of 10. • Expand whole numbers and decimals with positive and negative powers of 10. • Write numbers in scientific notation. • Identify coefficient and exponent in standard form. • Compare quantities written in scientific notation.
Our place-value system is built from powers of 10. Thousands, hundreds, tens and ones correspond to 10³, 10², 10¹ and 10⁰. Tenths, hundredths and thousandths correspond to 10⁻¹, 10⁻² and 10⁻³.
Problem
Write 47,561 using powers of 10.
- 1.4 is in the 10⁴ place, 7 in the 10³ place, 5 in the 10² place, 6 in the 10¹ place and 1 in the 10⁰ place.
- 2.47,561 = 4×10⁴ + 7×10³ + 5×10² + 6×10¹ + 1×10⁰.
Problem
Write 561.903 using powers of 10.
- 1.Use positive powers to the left of the decimal point and negative powers to the right.
- 2.561.903 = 5×10² + 6×10¹ + 1×10⁰ + 9×10⁻¹ + 0×10⁻² + 3×10⁻³.
Scientific Notation
Scientific notation writes a number as a coefficient multiplied by a power of 10. The coefficient must be at least 1 and less than 10.
For example, 5900 can be written as 590 × 10¹, 59 × 10², 5.9 × 10³ or 0.59 × 10⁴. Only 5.9 × 10³ is in scientific notation because its coefficient lies between 1 and 10.
Problem
Write 20,800 in scientific notation.
- 1.Move the decimal after the first non-zero digit: 2.08.
- 2.The scale needed is 10⁴.
- 3.So 20,800 = 2.08 × 10⁴.
Problem
Write 80,00,000 in scientific notation.
- 1.80,00,000 = 8,000,000.
- 2.This is 8 × 10⁶.
- 3.The coefficient 8 is already between 1 and 10.
When positive numbers have different exponents, compare the powers of 10 first. If the exponents match, compare the coefficients. The chapter also uses the number of digits in the coefficient to communicate how precisely an estimate is being stated.
Quiz
Which is in scientific notation?
What does 10⁻³ represent?
Write 65,950 in scientific notation.
Which is larger?
What is the coefficient in 3.081×10⁸?
Practice Problems
- Write 6,374 as an expanded sum using powers of 10.
- Write 42.507 using positive, zero and negative powers of 10.
- Convert 59,853 and 34,30,000 into scientific notation.
- Compare 1.4335×10¹² and 1.496×10¹¹ without expanding them fully.
- Write three different power-of-10 forms of 72,000 and identify the scientific-notation form.
Key Takeaways
• Place value is naturally described using powers of 10. • Negative powers of 10 describe decimal places. • Scientific notation has the form x×10ʸ with 1≤x<10. • The exponent indicates scale. • When exponents match, compare coefficients. • Scientific notation is useful for very large quantities and approximate measurements.