10.2. Powers with Negative Exponents

Exponent is a negative integer. You know that, 10² = 10 × 10 = 100.

In the case of negative exponents, the base remains the same, but the value is derived differently as shown below:

• 10⁰ = 1
• 10ˉ¹ = 1/10
• 10ˉ² = 1/(10 × 10) = 1/100
• 10ˉ³ = 1/(10 × 10 × 10) = 1/1000

Negative exponents represent the reciprocal of the base raised to the positive exponent. For example, 10⁻¹ means 1 divided by 10 raised to the power of 1, which is 1/10. Similarly, 10⁻² means 1 divided by 10², which results in 1/100. This pattern continues as the exponent decreases.

Think of a person's savings account. If they add money (positive exponent), their balance increases. If they withdraw money (negative exponent), you can think of it as moving the decimal point to the left, representing less money in the account.

Continuing the above pattern:

• 10ˉ¹ = 1 ÷ 10 = 0.1
• 10ˉ² = 1 ÷ (10 × 10) = 1 ÷ 100 = 0.01
• 10ˉ³ = 1 ÷ (10 × 10 × 10) = 1 ÷ 1000 = 0.001

Each negative exponent represents how many times we divide by the base number. For instance, 10⁻³ means we have divided by 10 three times, which is equivalent to finding how small a number gets as we keep dividing by 10.

Imagine cutting a chocolate bar. If you divide it into ten pieces, you have 1/10 of the bar. If you keep dividing each piece in the same manner, every time you divide, you're creating smaller pieces similar to how negative exponents work.

For example:

• 3³ = 3 × 3 × 3 = 27
• 3² = 3 × 3 = 9 = 27 ÷ 3 = 9
• 3¹ = 3 = 9 ÷ 3 = 3
• 3⁰ = 1 = 3/3 = 1
• 3ˉ¹ = 1 ÷ 3 = 1/3
• 3ˉ² = 1 ÷ (3 × 3) = 1/9
• 3ˉ³ = 1 ÷ (3 × 3 × 3) = 1/27.

By tracking the values of positive powers of 3, we can see how the values reduce as we step down towards negative exponents. Specifically, we see that as the exponent decreases from positive to negative, the values become fractions of a whole.

Consider a tree growing taller every year (positive powers) until one day it is cut down. Each year represents an increase, while moving backward portrays what happens to that tree as it loses size, just like how negative exponents depict a decrease in value.

In general, we can say that for any non-zero integer 'a', aˉm = 1/aᵐ, where m is a positive integer. aˉm is the multiplicative inverse of aᵐ.

This statement means that whenever you encounter a negative exponent, you can simply write it as 1 divided by the base raised to the opposite positive power. This inverses the value, hence the term multiplicative inverse.

Think of a seesaw. When one side (positive exponent) goes up, the other side (negative exponent) goes down and vice versa. This interplay helps us understand the balance between positive and negative values.

For instance:

• 10ˉ₂ = 1 ÷ 10² = 1/100 = 0.01
• 10ˉ₃ = 1 ÷ 10³ = 1/1000 = 0.001
• Likewise, 3ˉ₂ = 1 ÷ (3 × 3) =~ 0.111 (approximately after calculation).

By calculating the negative exponents, we are able to bridge the concept effectively with practical values. The decimal values highlight exactly how small these fractions become as exponent decreases.

Imagine pouring a whole jug of water out into smaller cups. The result is fractions of a whole jug, mimicking how negative exponents divide a full base into smaller pieces.

--