Express the power notation with positive exponent

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

What is power notation with positive exponent?

Power notations are basically the notations of exponents on a number or expression, the notation can be a positive or a negative term. Examples of the positive terms in exponent are 5⁶, x⁷, 8^p, etc. Examples of negative terms in exponent are 2^-3, 6^-7, x^-y, etc. Can the negative terms in exponent be represented as positive terms? YES. Using the rules of exponents, it is possible to represent the power notation with a positive exponent.

Let’s take an example of power notation with negative exponent and convert it into positive exponent, the term provided is 7^-x. -x is the exponent of the base 7, x has a negative sign assigned to it, in order to remove it, use the negative exponent rule which says, a^-m = 1/a^m where m is the exponent of a.

Applying the same on the term given, 

7^-x = 1/7^x

Therefore, the positive exponent is now 1/7^x.

Sample Problems

Question 1: Simplify and express the term 8¹ × 8³ in power notation with a positive exponent.

Solution:

Given term ⇢ 8¹ × 8³

Using the Product rule of exponents,

 8¹ × 8³ = 8^{(1 + 3)}

= 8⁴

Question 2: Simplify and express the term 3³ × 3^-5 in power notation with a positive exponent.

Solution:

Given term ⇢ 3³ × 3^-5

Using the Product rule of exponents,

3³ × 3^-5 = 3^{(3 + (-5))}

= 3^-2

= 1/3²

Question 3: Simplify and express the term -4¹⁰ ÷ -4⁵ in power notation with a positive exponent.

Solution

Given term ⇢ -4¹⁰ ÷ -4⁵ 

Using the Quotient rule of exponents,

-4¹⁰ ÷ -4⁵ = -4^{(10 – 5)}

= -4⁵