Exponents

Introduction

Talking about the “exponents” definition in detail, it describes the process of multiplying the same thing repeatedly. In (5)(5)(5) is equal to 53, for example, the shorthand for Xing three copies of the number 5 is displayed on the right-hand side of the “equals” sign. In this case, the “exponent,” 3, represents the number of times the item is multiplied. The “base” is the thing that is being multiplied, which is 5.

A number’s exponent indicates how many times the number has been multiplied by itself. 2 can be expressed as 24, for example, because 2 is multiplied by itself four times. 2 is referred to as the “base,” and 4 is referred to as the “exponent” or “power.” xn denotes that x has been multiplied by itself n times.

Properties of Exponents

Exponent characteristics, often known as exponent laws, solve problems involving exponents. These properties are also known as major exponents rules, which must be obeyed while dealing with exponents. An exponent of a number indicates how many times a number is multiplied by itself. For example, 34 denotes a four-fold multiplication of three. 3×3×3×3 is its expanded form. The power of a number is also known as the exponent. It can be any number, including whole numbers, fractions, negative numbers, and decimals.

Exponent qualities are discussed further down.

• Law of Product: am × an is equal to am+n
• Law of Quotient: am/an is equal to am-n
• Law of Zero Exponent: a0 is equal to 1
• Law of Negative Exponent: a-m is equal to 1/am
• Law of Power of a Power: (am)n is equal to amn
• Law of Power of a Product: (ab)m is equal to ambm
• Law of Power of a Quotient: (a/b)m is equal to am/bm

Negative Exponents

A negative exponent indicates how many times the reciprocal of the base must be multiplied. For instance, if a-n is supplied, it can be expanded to 1/an. It indicates we have to multiply 1/a ‘n’ times the reciprocal of a. When writing fractions using exponents, negative exponents are used.

Exponents with Fractions

A fractional exponent is when the exponent of a number is a fraction. Fractional exponents include square roots, cube roots, and the nth root. The square root of the starting point is a number having a power of 1/2. Similarly, the cube root of the base is an integer with a power of 1/3. Exponents with fractions include 52/3, -81/3, 105/6, and so on.

Decimal Exponents

A decimal exponent is one in which the exponent of a number is expressed in decimal form. Because evaluating the correct answer of any decimal exponent is a little challenging, we use an approximation in such circumstances. To solve decimal exponents, convert the decimal to fraction form first. 41.5, for example, can be expressed as 43/2, which can be simplified even further to achieve the final result of 8.

Scientific Notation with Exponents

The traditional form of writing is very big or very small numbers in scientific notation. In this, decimals and powers of ten are used to write numbers. For example, when a number from 0 to 9 is multiplied by a power of ten, it is considered to be written in scientific notation. When a number is larger than one, the power of ten is a positive exponent; when a number is less than one, the power of ten is negative.

To learn how to write numbers in scientific notation with exponents, follow these steps:

Step 1: After the first digit of the left-hand number, add a decimal point. We don’t need to put a decimal if there is only one digit in a number, omitting zeros.

Step 2: Multiply that number by 10 so that the power equals the number of times the decimal point is shifted.

Tips and Tricks:

• If a fraction does indeed have a negative exponent, the exponent is made positive by taking the reciprocal of the fraction. (a/b)-m = (b/a)m.
• We can set the bases equal when the exponents are equal, and vice versa. i.e., am = an ⇔ m = n.

We can write any number in standard form with exponents by following these two simple procedures, for example, 560000 = 5.6 × 105, 0.00736567 = 7.36567 × 10-3.

Conclusion

After learning about the exponent’s definition in detail, it could easily be said that exponents are useful in maths because they allow us to shorten something that would otherwise be extremely difficult to write. Moreover, exponents are a valuable resource. They’re used to demonstrate multiple multiplications. The exponent is the number of times a number is multiplied by itself. When rewriting an expression with exponents, however, we must pay attention to the placement of negatives and parentheses.