Negative Exponents: Rules, Fractions, and Calculations

What are Negative Exponents?

Negative exponents tell us how many times to divide the base number or, in other words, how many times to multiply the reciprocal of the base. To understand what are negative exponents, it is important to understand that a positive exponent shows repeated multiplication, while a negative exponent shows repeated division by the base.

For example, 4⁻³ = 1/4 × 1/4 × 1/4 = 1/64. Learning negative exponents is very important in maths because it helps solve algebra problems, work with scientific notation, and simplify expressions with smaller numbers easily. So, keep reading to learn more about negative exponent rules.

Read more: Perfect Cube

Negative Exponent Rules

Negative exponents follow some simple rules that make solving them easy. Here are the key negative exponent rules:

• Negative Exponent Rule 1: If a base number has a negative exponent, take the reciprocal of the base and then raise it to the positive exponent. For example, 4⁻³: 1/4³ = 1/4 × 1/4 × 1/4 = 1/64.

• Negative Exponent Rule 2: If a negative exponent is in the denominator, move the base to the numerator and make the exponent positive. For example, 1/2⁻³: 2³ = 2 × 2 × 2 = 8.

Read more: (a + b)³ Formula

Negative Fraction Exponents

In exams, questions based on negative fraction exponents are generally asked, like 4⁻³/². To solve this, you first need to use the rule for negative exponents: take the reciprocal of the base and make the exponent positive.

• For example: 4⁻³/² = 1 / 4³/²

• Now, simplify the fraction exponent using regular exponent rules. We know 4³/² = (2²)³/² = 2³ = 8.

• So, 4⁻³/² = 1 / 8.

This explains that negative fraction exponents combine both the negative exponent rule and fraction exponent rule to make calculations simple.

Read more: Surface Area of A Cube

Multiplying Negative Exponents

Multiplying negative exponents is done in the same way as multiplying positive exponents, but first we need to convert the negative exponents into fractions.

Let's understand it using an example: (4/5) ⁻³ × (10/3) ⁻²

• Step 1: Take the reciprocal of each base to make the exponents positive: (4/5)⁻³ × (10/3)⁻² = (5/4)³ × (3/10)²

• Step 2: Multiply the numbers using normal rules: (5/4)³ × (3/10)² = (5³ × 3²) / (4³ × 10²) = 125 × 9 / 64 × 100 = 1125 / 6400

So, the answer of multiplying negative exponents is 1125 / 6400.

This explains that multiplying negative exponents is easy if you first convert them to positive exponents using the reciprocal of the base.

Read more: Collinear Points

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How to Solve Negative Exponents?

In order to solve negative exponents, it is important to first convert them into positive exponents using these rules:

• a⁻ⁿ = 1 / aⁿ

• 1 / a⁻ⁿ = aⁿ

After this, we can simplify the expression using normal exponent rules.

Example: Solve (7³) × (3⁻⁴ / 21⁻²)

Step 1: Convert negative exponents to positive: (7³) × (3⁻⁴ / 21⁻²) = 7³ × (21² / 3⁴)

Step 2: Split 21² as (7 × 3)² = 7² × 3²: 7³ × (7² × 3² / 3⁴)

Step 3: Combine common bases using aᵐ × aⁿ = aᵐ⁺ⁿ and simplify 3² / 3⁴ = 3⁻² = 1 / 3²: 7³ × 7² × 3² / 3⁴ = 7⁵ / 3² = 16807 / 9.

Also read: Substitution Method

Negative Exponents Examples

Here are some solved examples to help you understand how negative exponents work in numbers and variables.

Example 1: Solve the expression (52 + 12)⁻²

Step 1: Add the numbers inside the bracket: 52 + 12 = 25 + 1 = 26
Step 2: Apply the negative exponent rule: (26)⁻² = 1 / 26²
Step 3: Multiply 26 × 26 = 676.

So, the answer is 1/676.

Example 2: Solve for x: 64 / 4⁻ˣ = 16.

Step 1: Write 64 and 16 as powers of 4: 64 = 4³, 16 = 4²
Step 2: Replace in the equation: 4³ / 4⁻ˣ = 4²
Step 3: Apply the rule aᵐ / aⁿ = aᵐ⁻ⁿ: 4³ × 4ˣ = 4² → 4³⁺ˣ = 4²
Step 4: Equate exponents: 3 + x = 2 → x = -1

So, the answer is x = -1.

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