Exponents Negative Numbers

Introduction:

         The exponent number is shows how many times to use the base number in a multiplication. The exponent number is placed at upper right of the base number. Exponent -x in the expression a^-x. For example, -4 is the exponent in 2^-4= 1/2⁴ = 0.0625. The base and exponent numbers may be positive or negative. Negative exponents are a way of indicating reciprocals.

Rules of Exponents Negative Numbers

Definitions

1. aⁿ = a·a·a···a  (n times)

2. a⁰ = 1  (a ≠ 0)

3. a^-1 = 1/aⁿ (a  ≠ 0)

4. a^m/n = ⁿ√a^m or (ⁿ√a)_m (a ≥ 0, m ≥ 0, n > 0)

Combining

1. Multiplication: a^x a^y = a^{x + y}

2. Division: a^x / a^y = a^x-y (a ≠ 0)

3. Powers: (a^x)^y = a^xy

Distributing   (a ≥ 0, b ≥ 0)

1. (ab)^x = a^x b^x

2. (a/b)^x = a^x/b^x  (b ≠ 0)

Careful!!

1. (a + b)ⁿ ≠ aⁿ + bⁿ

2. (a – b)ⁿ ≠ aⁿ – bⁿ

Rule for Exponents Negative Numbers:

a^-n = 1/aⁿ

Examples: 5^-2 = 1/5² = 1/25 

(2/3)^-3 = (3/2)³ = 27/8

Exponents Negative Numbers – Examples

Negative exponents numbers solved problems

Example 1: Solve this expression 8^-2

Solution:

     Here, the exponent is -2 negative exponents. Usually in positive exponent the exponent number is shows how many times to use the base number in a multiplication. In negative exponent number also shows like this. But in the negative exponent numbers we have to find the reciprocal of the numbers.

8^-2 = 1/8² = 1 / (8 × 8) = 1/64 = 0.015625

8² = 8 × 8 = 64.

Reciprocal of the number is 1/64 = 0.015625

Example 2: Solve 4^-3

Solution:

4^-3 = 1/4³ = 1 / (4 × 4 × 4) = 1/64 = 0.015625

Example 3: Solve this expression 2²/2^-3

Solution:

2²/2^-3 = 2² × 2³ = 2⁵ = 2 × 2 × 2 ×2 × 2 = 32

Example 4: Solve this expression 2(3^-1)

Solution:

2(3^-1) = 2(1/3) = 2/3 = 0.67

Example 5: Simplify this equation and solve this equation (4x)^-3, x = 2.

Solution:

(4x)^-3 = 1/64x³

Put x = 2

1/64(2³) = 1/64(8) = 1/512 = 0.001953125

Example 6: Simplify this equation and solve this equation (x^-3/y^-4)^-3, x = 1, y = 1.

Solution:

(x^-3 / y^-4)^-3 = (x^-3)^-3/(y^-4)^-3 = (y^-4)³/(x^-3)³ = y^-12/x^-9 = x⁹/y¹²

Put x =1 and y = 1 in the equation to get

1⁹/1¹² = 1/1 =1

Exponents Negative Numbers – Practice

Solve these problems for practice on negative exponents.

Problem 1: Solve 2^-2 - Answer: 0.25

Problem 2: Solve 3^-2 - Answer: 0.11

Problem 3: Solve 3²/3^-3 - Answer: 243

Problem 4: Solve this expression 4(4^-1) - Answer: 1

Problem 5: Simplify this equation and solve this equation (2x)^-2, x = 3. - Answer: 36