## How to Solve Zero and Negative Exponents

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We can tell how many times we are multiplying a number by itself by the exponent of a number. For example, the number \(3^3\) indicates that we are multiplying \(3\) three times. Its enlarged form is \(3 \times 3 \times 3\). The **exponent **of a number is often referred to as the **power **of a number. Whole numbers, fractions, negative numbers, and decimals can all be expressed in terms of exponents. In this lesson, we'll study a little more about exponents.

If you multiply a number by itself, the exponent of the number indicates how many times the number has been multiplied. For example, \(2 \times 2 \times 2 \times 2\) can be represented as \(2^4\), because 2 is **multiplied **by itself **four times**. In the term \(x^n\):

- \(n\) is the
**exponent**or power - \(x\) is called the
**base**

### Properties of Exponents

When dealing with exponents, it is **necessary **to apply the properties of exponents or the rules of exponents in order to solve the problem. These characteristics are also referred to as **major **exponents rules, which must be observed when solving exponent problems. The next section discusses the properties of exponents.

- The
**Product**Law states that \(a^m \times a^n \ = \ a^{m \ + \ n}\) - The Law of
**Quotients**states that \(\frac{a^m}{a^n} \ = \ a^{m \ - \ n}\) - The Law of the
**Zero**Exponent is as follows: \(a^0 \ = \ 1\) - The Law of the
**Negative**Exponent states that \(a^{-m} \ = \ \frac{1}{a^m}\) - The Law of the
**Power**of a**Power**is as follows: \((a^m)^n \ = \ a^{m \times n}\) - The Power of a
**Product**is defined as \((ab)^m \ = \ a^m \times b^m\) - The Power of a
**Quotient**is defined as \((\frac{a}{b} \ )^m \ = \ \frac{a^m}{b^m}\)

### Negative Exponents

A negative exponent tells us how many times we must multiply the reciprocal of the base in order to get the result we want. In the case of the expression \(a^{-n}\), it can be stretched to the expression \(\frac{1}{a^n}\). It implies that we must multiply the reciprocal of a, i.e., \(\frac{1}{a}\) '\(n\)' times, in order to get the answer. When writing fractions with exponents, it is **necessary **to utilise negative exponents. Examples of negative exponents include \(3^{-9}, \ 7^{-3}, \ 67^{-5}\), etc.

### Zero Exponents

Anything (a real number) raised to the power zero would **always **give the result as one. So, this means \(a^0 \ = \ 1.\)

### Exercises for Zero and Negative Exponents

**1) **\(-2^{-5} = \)

**2) **\(5^{-3} = \)

**3) **\(0^{5} = \)

**4) **\(6^{-5} = \)

**5) **\(-9^{-3} = \)

**6) **\(10^{-2} = \)

**7) **\(8^3\ \times \ 8 = \)

**8) **\(-8^{-2} = \)

**9) **\(0^{6} = \)

**10) **\(7^{-3} = \)