Key to Laws of Exponents

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10,000,000,000,000,000,000, it is not easy to read a number this big. Also, it is difficult to recognise and evaluate. To simplify this process, exponents are used. They help to assess and identify large numbers. Moreover, they are also known as powers or indices. A key to the laws of exponents can help students to get a grip on this subject.


What is an Exponent?

Exponents or powers indicate the times one number needs to be multiplied by itself. For instance, 43 means four needs to be multiplied thrice. So, instead of that, writing power is a concise way to represent this factor.


Here, 4 is the base and 3 is the power or exponent, also known as the index. When used in mathematics equations, these exponents follow specific laws, and a key to the laws of exponents can help in making the equations easy to solve and comprehend.


Laws of Exponent

Here are 6 laws of exponent with examples that can help students to comprehend this topic further:


1. Product of Power or Product Law

This law of exponent suggests that, while multiplying two numbers, where the base is the same, one can add its exponents. Product law of exponents examples is 43X45= 48


2. Quotient of Power or Quotient Law

When dividing two numbers with the same power or exponent, one can subtract the exponent values to complete this process. Quotient laws of exponents with examples is 48/44= 44.


3. Power of Power or Power Law

This law of exponent helps students to solve equations when power has another power. Power to power rule examples includes (53)5, here one need to multiply the powers, keeping the base unchanged.

Therefore, (53)5 = 515.


4. Power of a Product Law

This rule suggests that when a base gets multiplied by any exponent, one can distribute the exponent to every part of the base. This law of exponents examples and solutions includes, (axb)c= acXbc.


Moreover, this rule is also applicable if there are exponents with a base as well, (a2b2)3 = a6b6.


5. Power of a Quotient Law

Quotient refers to the division of two quantities. Here, similar to the previous one, students need to divide the exponents between all the values of a number. This exponent rule example is, (a/b)n = an/bn.


6. Zero Power Law

Zero power-law helps students to learn that when a number carries an exponent that is zero, it is equal to one. It means a0= 1. In any case, irrespective of how big or complicated the equation is, ‘zero’ exponent only results in one.


A key to the laws of exponent contains two more principals, which talks about negative and fractional exponent.

  • Negative Power Law

When power represents a negative value, one can simplify using the laws of exponents. In this process, using a reciprocal, one can turn it into a positive one.


For instance, x-a= 1/xa.

  • Fractional Power Law

Fractional exponents are also known as rational exponents. Here, among the digits of power, the numerator should be the number, and the denominator is the root.


For instance, a1/k= k√a.


These are some of the primary rules of exponents with examples. It is an essential concept of mathematics and crucial for exam preparations. Learning them thoroughly will help students to solve equations without any hiccups.


In this preparation process, a detailed explanation of a key to the laws of exponent is vital. Students can avail such important study materials from the website of Vedantu – India’s leading e-learning platform. Along with such study materials, students can also register for the live online classes, and doubt clearing sessions to improve their preparations further.

FAQ (Frequently Asked Questions)

1. How to Calculate Exponent?

Ans. Calculating exponents is not easy; one can simply multiply the base as many as many times indicated by the power. There are different processes to perform various equations associated with this topic. One can add the powers of two similar bases to simplify this process further. Contrarily one can subtract exponents when the bases are dividing each other. Apart from that, there are ways to calculate the power of a power and using negative exponents as a positive one to avoid complications in the calculation.

2. Why is Exponent Necessary?

Ans. It is difficult to read, analyse and calculate big numbers. In that case, exponents prove to be very useful. It helps to simplify this process of calculating, reading and analysing numbers. For instance, 3486784401 is a hard number to memorise, but if it is in form 910, then it is not hard to remember it. Moreover, in the case of an equation, this method accounts for a swift and error-free process.

3. What are the Laws of Exponents?

Ans. The laws of exponents include the product of power, the quotient of power, power of power, power of quotient, power of a product, zero power, negative and fractional power. Now, every law has a different purpose, and each of them solves a different purpose. Knowing them can help individuals during large calculations, and saves time.