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Negative Exponents

The exponents are known as powers which are raised to a variable or a constant. The expressions containing exponents are known as exponential expressions, such as: $x^{2}$.
Negative Exponents
Some exponential expressions have negative powers. These are known as negative exponents. For example: $a^{-3}$, $x^{-2}y^{-4}$.

Exponent represents the number of times the number is multiplied to itself, i.e.
$x^{y}=x \times x \times x \times .... x (y\ times)$

Here, y is a positive exponent of base x.

Exponential expressions with negative exponents are reversal to its positive exponent.

$x^{-y}=$$\frac {1}{x^{y}}$
Where -y is the negative exponent over x.

For example:

  • $x^{-2}=$$\frac{1}{x^{2}}$
  • $2^{-3}=$$\frac{1}{2^{3}}$

Negative exponents also follow all the other rules followed by positive exponents which are illustrated below:

  • $a^{m}\times b^{n}=a^{m+n}$
  • $\frac{a^{m}}{b^{n}}$$=a^{m-n}$
  • $(a^{m})^{n}=a^{mn}$
  • $a^{m}\times b^{m}=(ab)^{m}$
  • $\frac{a^{m}}{b^{m}}=(\frac{a}{b})^{m}$
  • $\frac{1}{a^{m}}$$=a^{-m}$

One must keep sign rules in mind while using above laws:

  • (Positive)(Positive) = Positive
  • (Negative)(Negative) = Positive
  • (Positive)(Negative) = Negative
  • (Negative)(Positive) = Negative

Let us consider few examples based on operation on negative exponents:

Example 1: Solve $\frac{4^{-2}}{2^{-4}}$.

Solution: $\frac{4^{-2}}{2^{-4}}$

= $\frac{\frac{1}{4^{2}}}{\frac{1}{2^{4}}}$

= $\frac{1}{4^{2}} \times \frac{2^{4}}{1}$

= $\frac{1}{16} \times \frac{16}{1}$

= 1

Example 2: Simplify $(3^{-2})^{(-2)}$
Solution: $(3^{-2})^{(-2)}$
= $3^{(-2) \times (-2)}$
= $3^{4}$
= 81

Related Calculators
Negative and Zero Exponents Calculator Calculating Exponents
Adding Negative Numbers

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