How to rewrite numbers without exponents?

Mathematics is not only about numbers but it is about dealing with different calculations involving numbers and variables. This is what basically is known as Algebra. Algebra is defined as the representation of calculations involving mathematical expressions that consist of numbers, operators, and variables. Numbers can be from 0 to 9, operators are the mathematical operators like +, -, ×, ÷, exponents, etc, variables like x, y, z, etc.

Exponents and Powers

Exponents and powers are the basic operators used in mathematical calculations, exponents are used to simplifying the complex calculations involving multiple self multiplications, self multiplications are basically numbers multiplied by themselves. For example, 7 × 7 × 7 × 7 × 7, can be simply written as 7⁵. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 11³, here, 11 is the base value and 3 is the exponent or power of 11. The value of 11³ is 1331.

Exponent is defined as the power given to a number, the number of times it is multiplied by itself. If an expression is written as cx^y where c is a constant, c will be the coefficient, x is the base and y is the exponent. If a number say p, is multiplied n times, n will be the exponent of p. It will be written as

p × p × p × p … n times = pⁿ

Basic rules of Exponents

There are certain basic rules defined for exponents in order to solve the exponential expressions along with the other mathematical operations, for example, if there are the product of two exponents, it can be simplified to make the calculation easier and is known as product rule, let’s look at some of the basic rules of exponents,

• Product Rule ⇢ aⁿ + a^m = a^{n + m}
• Quotient Rule ⇢ aⁿ / a^m = a^{n – m}
• Power Rule ⇢ (aⁿ)^m = a^{n × m} or ^m√aⁿ = a^n/m
• Negative Exponent Rule ⇢ a^-m = 1/a^m
• Zero Rule ⇢ a⁰ = 1
• One Rule ⇢ a¹ = a

How to rewrite numbers without exponents?

Expressing numbers or any other expression without exponents simply means to expand the expression or numbers and remove the symbol of exponent/power from it, simply removing the exponent will end up giving a wrong result, while removing the exponent it is important to address what is the power and on which number the power is located so as to expand it properly leading in the correct result or number.

Let’s take a look at a very simple example, 5⁴  where 4 is the exponent and 5 is the base, directly removing the exponent and writing 5 will provide the wrong result. 5⁴ means that 5 is multiplied by itself 4 times, therefore, after removing the exponent, it is important to expand the expression and rewrite it as 5 × 5 × 5 × 5 which is equal to 625.

Take another example, 7.89 × 10³, where 3 is the exponent of 10, and 7.89 is a decimal number. In order to remove the exponent 3, it is important to expand the base 10 and multiply it 3 times, hence it becomes 7.89 × 10 × 10 × 10 = 7.89 × 1000. The answer can be shown more neatly by simply multiplying 1000 with 7.89 and removing the decimal terms, hence, the answer is 7890.

Sample Problems

Question 1: Express the number 6.66 × 10² without exponent.

Solution:

To remove the exponent 2, expand 10 by multiplying it 2 times.

= 6.66 × 10 × 10

= 6.66 × 100

= 666.

Question 2: Express the number 23.897 × 10⁶ without exponent.

Solution:

To remove the exponent 6, expand 10 by multiplying it 6 times.

= 23.897 × 10 × 10 × 10 × 10 × 10 × 10

= 23.897 × 1000000

= 23897000.

Question 3: Express the term 7a²v⁵ without exponent.

Solution:

To remove the exponent 2, expand a by multiplying it 2 times and to remove 5, multiply v 5 times by itself.

= 7 × a × a × v × v × v × v × v.