## How to solve negative exponents and negative bases

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To understand more about negative exponents and negative bases, we need to first learn more about what exponents really are and their properties. So, as far as exponents are considered, they are basically the **multiplication **of a number by itself, a certain number of times. For example, let us consider the exponent \(4^8\). Here, we can see clearly that \(4\) is multiplied by itself \(8\) times. So, this can be written as \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\). So, this is why we resort to the exponential form as the general form can be very **tedious **and **exhaustive **to write.

Now for the exponent \(4^8\), we can see two parts, \(4\) and \(8\). In this case, \(4\) is referred to as the "**base**," while \(8\) is referred to as the "**exponent**" or "**power**." In the term \(x^n\):

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

Now, let’s understand about the properties of exponents.

### 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 and Negative Bases

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.

So, here \(-5x^{-2}\) and \((-5x)^{-2}\) are **not same**, the negative exponent only applies to \(x\). Always consider the **brackets, ** For example, \(2x^{-3}\) would be written as \(\frac{2}{x^3}\) .

### Exercises for Negative Exponents and Negative Bases

**1) **\(-\frac{7x}{x^{-5}} = \)

**2) **\(-\frac{14x}{x^{-5}} = \)

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

**4) **\(-\frac{16}{x^{-4}} = \)

**5) **\(\frac{11x^2}{-3y^{-5}} = \)

**6) **\(\frac{18x^4}{-3y^{-2}} = \)

**7) **\((\frac{3}{4})^{-2} = \)

**8) **\((\frac{-5}{2})^{-2} = \)

**9) **\((\frac{-4}{2})^{-2} = \)

**10) **\(\frac{19x^3}{-2y^{-5}} = \)