How to Solve Zero and Negative Exponents

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 \(a^0 \ = \ 1.\)

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