Understanding powers of ten
Powers of ten are a way of writing numbers with exponents, like 10², which is 100. They are the building blocks of our entire number system and a shortcut for writing very large (or very small) numbers. 🚀
Concept
Powers of ten are numbers that can be written as 10 raised to an exponent. The exponent simply tells you how many times to multiply 10 by itself, or more easily, how many zeros to write after the 1. **Understanding the Exponent:** * 10¹ = 10 (10 to the power of 1) * 10² = 10 x 10 = 100 (10 to the power of 2, or "10 squared") * 10³ = 10 x 10 x 10 = 1,000 (10 to the power of 3, or "10 cubed") * 10⁴ = 10 x 10 x 10 x 10 = 10,000 * ...and so on! **The Zero Trick:** The exponent tells you exactly how many zeros follow the 1. * 10⁵ = 100,000 (5 zeros) * 10⁶ = 1,000,000 (6 zeros - one million!) **Multiplying with Powers of Ten:** When you multiply a number by a power of ten, you are making the number larger by shifting the decimal point to the right. The exponent tells you how many places to shift. * **8.5 x 10² (or 8.5 x 100):** * The exponent is 2, so move the decimal 2 places to the right. * 8.5 -> 85. -> 850. * Answer: **850** * **0.43 x 10³ (or 0.43 x 1,000):** * The exponent is 3, so move the decimal 3 places to the right. * 0.43 -> 4.3 -> 43. -> 430. * Answer: **430** **Dividing with Powers of Ten:** When you divide, you do the opposite: move the decimal point to the left. * **720 ÷ 10² (or 720 ÷ 100):** * The exponent is 2, so move the decimal 2 places to the left. * 720. -> 72.0 -> 7.20 * Answer: **7.2** **Key Idea:** Powers of ten are directly linked to place value. Each place (tens, hundreds, thousands) is a power of ten. 4,000 is the same as 4 x 1,000, or 4 x 10³.
Try it
Practice with powers of ten! **Evaluate the Power:** 1. 10³ = ? 2. 10⁵ = ? 3. 10¹ = ? 4. 10⁶ = ? **Multiply by Powers of Ten:** 5. 6.7 x 10² = ? 6. 0.123 x 10³ = ? 7. 45 x 10⁴ = ? 8. 9.0 x 10¹ = ? **Divide by Powers of Ten:** 9. 800 ÷ 10² = ? 10. 4,500 ÷ 10³ = ? 11. 67.8 ÷ 10¹ = ? **Word Problems:** 12. A company made a profit of 4.5 million dollars. Write this number as 4.5 multiplied by a power of ten. 13. The distance to the moon is about 2.4 x 10⁵ miles. Write this as a standard number. 14. A computer can do a calculation in 3 x 10⁻⁹ seconds (a billionth of a second!). This is a tiny number. While we focus on positive exponents, this shows how powers of ten can represent small values too. **Challenge Problems:** 15. What is 52 multiplied by 10⁵? 16. If you divide 7,800 by 10³, what is the result?