Negative Exponents

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Thickness of a piece of paper is 0.00016 m
The size of a plant cell is 0.00001275 m.
Diameter of a wire on a computer chip is 0.000003 m.
The average diameter of a Red Blood Cell is 0.000007 m.

Arrange these in ascending order of size ?
We know how to write large numbers more conveniently using exponents. Can we write these very small numbers in exponential form?

Powers with Negative Exponent

↬10² = 10×10 =100
   10¹ = 10
   10⁰ = 1
   10^-1 = ?
As the exponent decreases by 1, the value becomes one-tenth of the previous value.
continuing the above pattern we get,
10⁻¹ = ¹/₁₀
10⁻² = ¹/₁₀ ÷ 10 = ¹/₁₀₀ = ¹/_10²
10⁻³ = ¹/₁₀₀ ÷10 = ¹/₁₀₀₀ =¹/_10³
Let's change the base and observe the pattern,
3² = 3×3 =9
3¹ = 3
3⁰ = 1
3⁻¹ = ?
As the exponent decreases by 1, the value becomes one-third of the previous value.
continuing the above pattern we get,
3⁻¹ = ¹/₃
3⁻² = ¹/₃ ÷ 3 = ¹/_3×3 =¹/_3²
3⁻³ = ¹/_3×3 ÷3 = ¹/_3×3×3 =¹/_3³

↬ In general, we can say that, for any non-zero integer a and m, 

a^−m =¹/_a^m ,            

We know that ¹/_a^m  is multiplicative inverse of a^m 
⇒ a^−m is the multiplicative inverse (reciprocal) of a^m.

↬ We can expand decimals using negative exponents as,
526.193 = 5 × 100 + 2 × 10 + 6 × 1 + ¹/₁₀  + ⁹/₁₀₀+ ³/₁₀₀₀
   = 5 × 10² + 2 × 10 + 6 × 1 + ¹/₁₀  + ⁹/_10² + ¹/_10³
   = 5 × 10² + 2 ×10¹ + 6 ×10⁰ + 1× 10^-1  + 9× 10⁻² + 3× 10⁻³

Laws of Exponents

Laws of exponents which hold for positive exponents also hold for negative exponents.
For any non-zero integers a and b and any integers m and n .

→ a0 = 1	exponent 0
→ am × an = a m+n	same base
→ am ÷ an = a m−n	same base
→ (am)n = a m×n	power of power
→ am × bm = (a×b)m	same exponent
→ am ÷ bm = (a/b)m	same exponent
→ (a/b)−m  = (b/a)m	negative exponent

Expressing very small number in Standard Index Form

→ Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10 . Such a form of a number is called its standard form (or standard index form).
→ a × 10ⁿ  (where a is any real number such that 1≤|a|<10 and n is any integer. )
e.g., 799 = 7.99× 10²
        79.9 = 7.9×10
        0.799 = 7.99× 10⁻²
        0.0799 =7.99×10⁻²
        0.00799 =7.99×10⁻³ & so on.

→ Very small numbers can be expressed in standard form using negative exponents.

→ Now we can express small number asked earlier in standard form
Thickness of a piece of paper is 0.000016 m = 1.6×10^-5 m
The size of a plant cell is 0.00001275 m = 1.275×10^-5 m
Diameter of a wire on a computer chip is 0.000003 m = 3×10^-6 m .
The average diameter of a Red Blood Cell is 0.000000007 m = 1.6×10^-9 m

Comparing numbers in Standard Index Form

To compare numbers in standard form, we convert them into numbers with the same exponents.
∎ Diameter of the Sun = 1.4 × 10⁹ m
     Diameter of the earth = 1.2756 × 10⁷ m
     Which is bigger and by how many times ?
↬ Diameter of the Sun = 1.4 × 10⁹ m
     = 1.47× 10²× 10⁷ = 147× 10⁷ m
     Diameter of the Earth = 1.2756 × 10⁷ m
     Now, the exponents are equal, we can compare them, 147 > 1.2756
     So, Sun is bigger than Earth by = 147× 10⁷ / 1.2756 × 10⁷ ≅ 100 times.

Adding/Subtracting numbers in Standard Form

∎ To add or subtract numbers in standard form, we convert them into numbers with the same exponents.
e.g., 1.496 × 10¹¹ – 3.84 × 10⁸
= 1496 × 10⁸ – 3.84 × 10⁸
= (1496 – 3.84) × 10⁸
= 1492.16 ×10⁸

Exponents & Powers
	Square & Square Roots ➤