Multiplying by Powers of Ten
Multiplying by powers of ten is a mental math shortcut in Grade 8 math (Yoshiwara Core Math) that moves the decimal point. Multiply by 10ⁿ: move the decimal n places right, inserting zeros as needed. For example, 0.017 × 10⁴ = 170; 28 × 10⁸ = 2,800,000,000. For negative exponents (× 10⁻³), move left. This is foundational for scientific notation, metric conversions, and estimation in science and engineering.
Key Concepts
Property To multiply a number by a power of ten: 1. Move the decimal point to the right the same number of places as the exponent on ten. 2. Write in zeros to fill any empty places at the end of the new number.
Examples To compute $56 \times 10^4$, move the decimal point 4 places to the right from $56.0$, giving $560,000$. To compute $4.81 \times 10^6$, move the decimal point 6 places to the right, which results in $4,810,000$. To compute $0.092 \times 10^3$, move the decimal point 3 places to the right, giving $92$.
Explanation This rule is a shortcut based on our base 10 system. Multiplying by $10^n$ means making the number $10^n$ times larger. This is visually represented by shifting the digits $n$ places to the left, which we accomplish by moving the decimal point right.
Common Questions
How do you multiply by 10⁴?
Move the decimal point 4 places right. 0.017 × 10⁴ = 170.
What happens when multiplying a whole number by 10⁸?
Append 8 zeros. 28 × 10⁸ = 2,800,000,000.
How does this relate to scientific notation?
Scientific notation writes (1–9.9) × 10ⁿ. Multiplying by powers of 10 shifts the decimal and adjusts the exponent.
What does multiplying by 10⁻³ mean?
Move the decimal 3 places left (divide by 1000). 5.2 × 10⁻³ = 0.0052.
How do powers of ten help metric conversions?
Metric units differ by powers of 10. Converting km to m multiplies by 10³.