Expand 102.563 Using Exponents

Number: 102.563

Identify each digit and its place: Digit → Place → Power of 10 1 → Hundreds → 10² 0 → Tens → 10¹ 2 → Ones → 10⁰ . → Decimal point 5 → Tenths → 10⁻¹ 6 → Hundredths → 10⁻² 3 → Thousandths → 10⁻³

Expanded form: 102.563 = 1×10² + 0×10¹ + 2×10⁰ + 5×10⁻¹ + 6×10⁻² + 3×10⁻³

Verification: = 100 + 0 + 2 + 0.5 + 0.06 + 0.003 = 102.563 ✓

Place → Power → Value Hundreds → 10² → 100 Tens → 10¹ → 10 Ones → 10⁰ → 1 Tenths → 10⁻¹ → 0.1 Hundredths → 10⁻² → 0.01 Thousandths → 10⁻³ → 0.001 Ten-thousandths → 10⁻⁴ → 0.0001

Key rules: • Digits left of decimal point use positive powers: 10⁰, 10¹, 10², ... • Digits right of decimal point use negative powers: 10⁻¹, 10⁻², 10⁻³, ... • 10⁰ = 1 (ones place) • 10⁻¹ = 1/10 = 0.1 (tenths place)

Example 1: Expand 45.72 = 4×10¹ + 5×10⁰ + 7×10⁻¹ + 2×10⁻² = 40 + 5 + 0.7 + 0.02 = 45.72 ✓

Example 2: Expand 306.018 = 3×10² + 0×10¹ + 6×10⁰ + 0×10⁻¹ + 1×10⁻² + 8×10⁻³ = 300 + 0 + 6 + 0 + 0.01 + 0.008 = 306.018 ✓

Example 3: Expand 0.075 = 0×10⁰ + 0×10⁻¹ + 7×10⁻² + 5×10⁻³ = 0 + 0 + 0.07 + 0.005 = 0.075 ✓

General method:

1. Identify the place value of each digit.
2. Write digit × 10^(place exponent).
3. Sum all terms.