To expand 102.563 using exponents, write each digit multiplied by the power of 10 that corresponds to its place value. 102.563 = 1×10² + 0×10¹ + 2×10⁰ + 5×10⁻¹ + 6×10⁻² + 3×10⁻³.
102.563 = 1×10² + 0×10¹ + 2×10⁰ + 5×10⁻¹ + 6×10⁻² + 3×10⁻³.
Digits left of decimal: positive powers of 10 (10⁰, 10¹, 10², ...).
Digits right of decimal: negative powers of 10 (10⁻¹, 10⁻², 10⁻³, ...).
10⁰ = 1; 10⁻¹ = 0.1; 10⁻² = 0.01; 10⁻³ = 0.001.
Verification: sum all terms to get back the original number.
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:
102.563 = 1×10² + 0×10¹ + 2×10⁰ + 5×10⁻¹ + 6×10⁻² + 3×10⁻³. Each digit is multiplied by 10 raised to the power of its place value position.
5 is in the tenths place. Its place value = 5 × 10⁻¹ = 0.5.
A negative exponent represents a fractional place value to the right of the decimal point. 10⁻¹ = 0.1 (tenths), 10⁻² = 0.01 (hundredths), 10⁻³ = 0.001 (thousandths).
45.72 = 4×10¹ + 5×10⁰ + 7×10⁻¹ + 2×10⁻² = 40 + 5 + 0.7 + 0.02.
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