Exponents

Exponent of a number shows how many times we are multiplying a number by itself. For example, 3⁴ means we are multiplying 3 four times. Its expanded form is 3×3×3×3. Exponent is also known as the power of a number. It can be a whole number, fraction, negative number, or decimals. Let's learn more about exponents in this lesson.

Table of Contents

• What are Exponents?
• Negative Exponents
• Fractional Exponents
• Decimal Exponents
• Scientific Notation with Exponents
• Laws (Properties or Rules) of Exponents
• FAQs on Exponents
• Solved Examples
• Practice Questions

The exponent of a number shows how many times the number is multiplied by itself. For example, 2×2×2×2 can be written as 2⁴, as 2 is multiplied by itself 4 times. Here, 2 is called the "base" and 4 is called the "exponent" or "power." In general, xⁿ means that x is multiplied by itself for n times.

Here, in the term xⁿ,

• x is called the "base"
• n is called the "exponent" or "power"
• xⁿ is read as "x to the power of n" (or) "x raised to n"

Examples of Exponents

Some examples of exponents are as follows:

• 3 × 3 × 3 × 3 × 3 = 3⁵
• -2 × -2 × -2 = (-2)³
• a × a × a × a × a × a = a⁶

Why are Exponents Important?

Exponents are important because, without them, the products where a number is repeated by itself many times is very difficult to write. For example, it is very easy to write 5⁷ instead of writing 5 × 5 × 5 × 5 × 5 × 5 × 5.

A negative exponent tells us how many times we have to multiply the reciprocal of the base. For example, if it is given that a^-n, it can be expanded as 1/aⁿ. It means we have to multiply the reciprocal of a, i.e 1/a 'n' times. Negative exponents are used to write fractions with exponents. Some of the examples of negative exponents are 2 × 3^-9, 7^-3, 67^-5, etc. To learn more about negative exponents, click here.

If an exponent of a number is a fraction, it is known as a fractional exponent. Square roots, cube roots, n^th root are all parts of fractional exponents. Number with power 1/2 is termed as the square root of the base. Similarly, numbers with power 1/3 is called cube root of the base. Some examples of fractional exponents are 5^2/3, -8^1/3, 10^5/6, etc. To learn more about fractional exponents, click here.

If an exponent of a number is given in the decimal form, it is known as a decimal exponent. It is slightly difficult to evaluate the correct answer of any decimal exponent so we find the approximate answer for such cases. Decimal exponents can be solved by first converting the decimal in fraction form. For example, 4^1.5 can be written as 4^3/2 which can be simplified further to get the final answer 8 or -8.

Scientific notation is the standard form of writing very large numbers or very small numbers. In this, numbers are written with the help of decimal and powers of 10. A number is said to be written in scientific notation when a number from 0 to 9 is multiplied by a power of 10. In the case of a number greater than 1, the power of 10 will be a positive exponent, while in the case of numbers less than 1, the power of 10 will be negative. Let's understand the steps for writing numbers in scientific notation:

• Step 1: Put a decimal point after the first digit of the number from the right. If there is only one digit in a number excluding zeros, then we don't need to put decimal.
• Step 2: Multiply that number with a power of 10 such that the power will be equal to the number of times we shift the decimal point.

By following these two simple steps we can write any number in standard form with exponents, for example, 560000 = 5.6 × 10⁵, 0.00736567 = 7.36567 × 10^-3.

To learn more about the use of exponents in writing scientific notation of numbers, visit the following articles:

• How Do You Write 2.5 million in Scientific Notation?
• How do you write 12 million in scientific notation?
• How do you write 0.0001 in scientific notation?
• What is the scientific notation for 8 million?
• How do you write 13 million in scientific notation?
• Which of the following expressions is written in scientific notation

The laws (properties or rules) of exponents are used to solve problems involving exponents. The laws of exponents are mentioned below.

• Law of Product: a^m × aⁿ = a^m+n
• Law of Quotient: a^m/aⁿ = a^m-n 
• Law of Zero Exponent: a⁰ = 1
• Law of Negative Exponent: a^-m = 1/a^m
• Law of Power of a Power: (a^m)ⁿ = a^mn 
• Law of Power of a Product: (ab)^m = a^mb^m
• Law of Power of a Quotient: (a/b)^m = a^m/b^m 

To learn more about exponent rules, click here.

Tips and Tricks:

• If a fraction has a negative exponent, then we take the reciprocal of the fraction to make the exponent positive. i.e., (a/b)^-m = (b/a)^m.
• When the exponents are the same, we can set the bases equal and vice versa. i.e., a^m = aⁿ ⇔ m = n.

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Example 1: The dimensions of a wardrobe are x⁵ units, y³ units, and x⁸ units. Find its volume.

Solution:

The dimensions of the wardrobe are length= x⁵ units, width= y³ units and heigth= x⁸ units. The volume of the wardrobe is, volume= lwh. So, by substituting the values, the volume of the wardrobe is x⁵ × y³ × x⁸= x¹³y³. Therefore, the volume of the wardrobe is x¹³y³ cubic units.

Example 2: In a forest, each tree has about 5⁷ leaves and there are about 5³ trees in the forest. Find the total number of leaves.

Solution:

The number of trees in the forest = 5³ and the number of leaves in each tree = 5⁷. The total number of leaves are: 5³ × 5⁷ = 5¹⁰ (because a^m × aⁿ = a^(m+n)). Therefore, the total number of leaves is 5¹⁰.

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.