L3 · Powers of 10 and place value

Learning goal

“I can express any power of 10 as a decimal number and as a fraction, read the exponent of a power of 10 as a place-value shift, and write whole numbers and small decimals as a single digit times a power of 10.”

Power of 10 — 10 raised to an integer exponent: 10⁴, 10⁰, 10⁻³. The exponent counts place-value steps from 1.

Place value — the value of a digit's position: in 5,203 the 2 sits in the hundreds place, worth 2 × 10².

Standard form — an ordinary decimal number as usually written: 100,000 or 0.001 (as opposed to the power-of-10 form 10⁵ or 10⁻³).

Magnitude — the size scale of a number, named by its power of 10: 8,500 has the magnitude of thousands (between 10³ and 10⁴).

Worked example

Give 10⁻² three faces. Step 1: As a power: 10⁻². Step 2: As a fraction: the negative exponent law gives 10⁻² = 1/10² = 1/100. Step 3: As a decimal: 1/100 = 0.01 - the hundredths place. Say it plainly: each step down the ladder divides by 10 and moves the decimal point one place left: 10² = 100, 10¹ = 10, 10⁰ = 1, 10⁻¹ = 0.1 = 1/10, 10⁻² = 0.01 = 1/100, 10⁻³ = 0.001 = 1/1000. Positive exponents count zeros after the 1; negative exponents count decimal places.

Check yourself — explain it

Why does 10⁻³ = 0.001 and not -1,000?

Paired problem — example still visible

Give 10⁴ and 10⁻¹ each as a decimal and (for 10⁻¹) as a fraction -> 10⁴ = 10,000; 10⁻¹ = 0.1 = 1/10.

Faded completion — last step is yours

"10⁻⁴ = 1/______ = ______" -> 10,000 (i.e. 1/10⁴); 0.0001

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