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Write The Expression with Positive Exponents Calculator

Reviewed by Calculator Editorial Team

This calculator helps you rewrite mathematical expressions with negative exponents to use only positive exponents. It follows the fundamental exponent rule that states any non-zero number raised to a negative exponent is equal to 1 divided by that number raised to the positive exponent.

What is Positive Exponents?

In mathematics, exponents indicate how many times a number (the base) is multiplied by itself. Positive exponents represent repeated multiplication, while negative exponents represent reciprocals. The key rule for converting negative exponents to positive exponents is:

Exponent Conversion Rule

For any non-zero number a and integer n:

a⁻ⁿ = 1 / aⁿ

This rule allows you to rewrite any expression with negative exponents using only positive exponents by moving the base to the denominator.

How to Convert Expressions

To convert an expression with negative exponents to one with positive exponents:

  1. Identify all terms with negative exponents in the expression.
  2. For each term with a negative exponent, apply the conversion rule: a⁻ⁿ = 1 / aⁿ.
  3. Combine all converted terms with the remaining positive exponent terms.
  4. Simplify the expression by combining like terms if possible.

Important Notes

  • The base a must not be zero, as division by zero is undefined.
  • This conversion works for all integer exponents, not just negative ones.
  • The order of operations (PEMDAS/BODMAS) should be followed when simplifying.

Worked Examples

Example 1: Simple Negative Exponent

Convert x⁻³ to positive exponents.

Using the conversion rule:

x⁻³ = 1 / x³

Example 2: Multiple Negative Exponents

Convert 2⁻² × 3⁻¹ to positive exponents.

Applying the rule to each term:

2⁻² = 1 / 2² and 3⁻¹ = 1 / 3¹

Combining them gives:

(1 / 2²) × (1 / 3¹) = 1 / (2² × 3¹) = 1 / 12

Example 3: Complex Expression

Convert 5⁻² × y⁴ × z⁻³ to positive exponents.

Applying the rule to the negative exponents:

5⁻² = 1 / 5² and z⁻³ = 1 / z³

Combining all terms gives:

(1 / 5²) × y⁴ × (1 / z³) = y⁴ / (5² × z³)

FAQ

Can I use this calculator for fractional exponents?

Yes, the same conversion rule applies to fractional exponents. For example, a⁻¹/² = 1 / a¹/².

What happens if the base is zero?

Zero raised to any negative exponent is undefined because division by zero is not allowed.

Is this conversion always valid?

Yes, the conversion is mathematically valid for all non-zero bases and integer exponents.

Can I use this for scientific notation?

Yes, the same rules apply to numbers in scientific notation, such as 1.23 × 10⁻⁵.