Exponents

An 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 article.

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

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

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

The properties of exponents or laws of exponents are used to solve problems involving exponents. These properties are also considered as major exponents rules to be followed while solving exponents. The properties 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

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 while writing fractions with exponents. Some of the examples of negative exponents are 2 × 3^-9, 7^-3, 67^-5, etc. We can convert these into positive exponents as follows:

• 2 × 3^-9 = 2 × (1/3⁹) = 2 / 3⁹
• 7^-3 = 1/7³
• 67^-5 = 1/67⁵

If an exponent of a number is a fraction, it is known as a fractional exponent. Square roots, cube roots, n^th root are parts of fractional exponents. Number with power 1/2 is termed as the square root of the base. Similarly, a number with a power of 1/3 is called the cube root of the base. Some examples of exponents with fractions are 5^2/3, -8^1/3, 10^5/6, etc. We can write these as:

• 5^2/3 = (5²)^1/3 = 25^1/3 = ∛25
• -8^1/3 = ((-2)³)^1/3 = -2
• 10^5/6 = (10⁵)⁶ = ⁶√10⁵ = ⁶√100000

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. i.e., 4^3/2 = (2²)^3/2 = 2³ = 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 between 0 to 10 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 with exponents:

• Step 1: Put a decimal point after the first digit of the number from the left. 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

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 in an equation on both sides, we can set the bases equal and vice versa. i.e., a^m = aⁿ ⇔ m = n.

☛ Related Topics on Exponents

Check a few more interesting articles based on the exponents in math.

• Multiplying Exponents
• Exponential Functions
• Exponential Equations
• Irrational Exponents

What are Exponents in Math?

An exponent is a number that is placed as a superscript over a number. In other words, it indicates that the base is raised to a certain power. The exponent is also called by other names like index and power. If m is a positive number and n is its exponent, then mⁿ means m is multiplied by itself n times.

What are the Properties of Exponents?

Laws of exponents are some rules that we use to do calculations involving exponents. These rules help us to calculate quickly. Laws of exponents or a few important properties of exponents are listed below:

• a^m × aⁿ = a^m+n
• a^m/aⁿ = a^m-n
• a⁰ = 1
• a^-m = 1/a^m
• (a^m)ⁿ = a^mn
• (ab)^m = a^mb^m
• (a/b)^m = a^m/b^m

What are the Examples of Exponents?

Some examples of exponents are as follows:

• 7 × 7 × 6 × 6 × 6 = 7² × 6³
• -4 × -4 × -4 × -4 = (-4)⁴
• p × p × p × p × p = p⁵

How do Exponents Relate to Real Life?

In real life, we use the concept of exponents to write numbers in a simplified manner and in a short way. Repeated multiplication can be easily written with the help of exponents. Also, we use exponents to write larger numbers, for example, the distance of the moon from earth, the number of bacteria present on a surface, etc.

How to Add Exponents?

Exponents cannot be added. We can only add like terms (terms having the same exponent and same variable). But, in the case of the multiplication of terms with the same variables, we add the exponents of the variable to multiply. For example, x² × x⁴ = x⁽²⁺⁴⁾ = x⁶. Try Cuemath's adding exponents calculator and get your answers quickly and easily.

Why are Exponents Important?

Exponents are important to write the values of numbers in simplified form. We know that repeated addition can be written as multiplication. In the same way, repeated multiplication can be written simply with the help of exponents.

How to Calculate Exponents Using Exponents Calculator?

An 'Exponent Calculator' is an online tool that finds the value for an exponential expression. Check now Cuemath's exponent calculator and find the value of an exponential expression for a given base and exponent value within a few seconds.

☛ Also Check:

• Negative Exponents Calculator
• Dividing Exponents Calculator
• Multiplying Exponents Calculator
• Exponent Rules Calculator
• Fractions with Exponents Calculator

How to Multiply Exponents?

When exponents are required to be multiplied, we first solve the numbers within the parenthesis, the power outside the parenthesis is multiplied with every power inside the parenthesis. For example, (3x²y³)² = 3² × x^2×2 × y^3×2 = 9x⁴y⁶.

What is the use of the Properties of Exponents?

There is a major use of properties of exponents in mathematics, especially in algebra. With the help of the properties of exponents, we can easily simplify the expressions and also write the expressions in fewer steps. Let us understand this with a simple example. With the help of exponents properties, 2⁴ × 2⁶ can be simplified in two quick steps as 2⁴ × 2⁶ = 2^{(4 + 6)} = 2¹⁰.

What are the Real-Life Applications of Exponents?

Exponents have various applications. Few exponents applications are listed below:

• Exponents are widely used in computer games, measuring scales, etc.
• Scientific scales like the pH scale or the Richter scale are based on exponents.
• They are used while calculating the area, volume, and problems related to measurement.
• They are most commonly used in the respective field of science, engineering, economics, accounting, and finance.
• They are often used to represent a computer's or laptop's memory.

How are Laws of Exponents Used in Algebra?

The laws of exponent are very useful in algebra. For example, the algebraic formula of (a - b)² = a² + b² - 2ab can be written and calculated easily by applying the rules of exponents. Many such algebraic formulas are dependent only on the laws of exponents.

How are Negative Exponents Used in Real Life?

Negative exponents are used in writing very small numbers in real life, which means numbers having values between 0 to 1.

What is a Zero Exponent?

Zero exponent means numbers that have 0 as their exponent. The values of those numbers are always 1. Any number with 0 as its exponent is equal to 1.