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 \(\frac{2}{x^3}\) .

Free printable Worksheets

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

Answers

1) \(-\frac{7x}{x^{-5}} = \)\( \ \color{red}{-7x^{6}}\)

2) \(-\frac{14x}{x^{-5}} = \)\( \ \color{red}{-14x^{6}}\)

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

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

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

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

7) \((\frac{3}{4})^{-2} = \)\( \ \color{red}{\frac{16}{9}}\)

8) \((\frac{-5}{2})^{-2} = \)\( \ \color{red}{\frac{4}{25}}\)

9) \((\frac{-4}{2})^{-2} = \)\( \ \color{red}{\frac{4}{16}}\)

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

How to solve negative exponents and negative bases