Simplify 8/5x-2/3

Algebra is the branch of mathematics that deals with unknown quantities by assigning them various letters, symbols, or alphabets, called variables. They are so-called because they are subject to change over a period of time with respect to whatever situation is given, but there is a constant need to represent these changing parameters. The most commonly used variables in algebra are the English alphabets in general and x, y, z in particular.  

Algebraic Expression

In simple words, such expressions as are formed with the help of algebra are called algebraic expressions. In mathematics, usually, a number is used to express a quantity. In some cases, though, numbers can be expressed using letters or alphabets, or other symbols without mentioning their actual quantities. Further, these individual quantities can be added, subtracted, raised to an exponent, radicalized, multiplied, or even divided to form some kind of expression relevant to the topic on hand.  

• 5b + 69 is an algebraic expression with b as the variable, 5 as the coefficient, 69 as the constant term and addition as the arithmetic operator.
• 420p² – 6900 is an algebraic expression with p as the variable, 420 as the coefficient, 6900 as the constant term and subtraction and exponent as the arithmetic operators.
• 24v² – 69vr + 420r – 78 is an algebraic expression with v and r as the variables, 24, 69, 420 as the coefficients, 78 as the constant term with addition, exponent, subtraction and multiplication as the arithmetic operators.

Laws of Exponents

There are different laws made for exponents to make the complex calculations easier, for instance, the exponents can be broken into two, if the two terms have the same base, they can be solved using one law, etc. Let’s take a look at these laws in more detail,

• Law 1 (Product Law): Where the same base is in multiplication with itself with distinct exponents, such exponents are added together, i.e., p^u × p^v = p^{u + v}.

Example:

1. 62³⁷ × 62⁵⁰ = 62^{37 + 50} = 62⁸⁷
2. 74^-12 × 74³² × 74¹⁰¹ = 74^{-12 + 32 + 101} = 74¹²¹

• Law 2 (Quotient law): Where the same base is in the division with itself with distinct exponents, such exponents are subtracted in the numerator, i.e., p^m ÷ pⁿ = p^m-n.

Example:

1. = 2^{40 – 5} = 2³⁵
2. = k^{4 – 12} = k^-8

• Law 3 (Zero exponent law): Where a base has been raised to the power of zero, its value is always 1, p⁰ = 1.

Example:

• 20⁰ = 1
• 100⁰ = 1
• 69420⁰ = 1

• Law 4 (Power law): Where the exponent is further raised to an exponent, both of the exponents are first multiplied and then further evaluation is done, i.e., (p^m)ⁿ = p^mn.

Example:

1. (112³⁰)⁴⁰ = (112)^{30 × 40} = 112¹²⁰⁰
2. [(-13)^-90]² = (-13)^{-90 × 2} = (-13)^-180

• Law 5: Where two different bases having equal exponents are in multiplication, their product gets raised to the given exponent, i.e., p^m × q^m = (p × q)^m.

Example:

1. 40³⁶ × 100³⁶ = (40 × 100)³⁶ = 4000³⁶
2. 20¹³ × 16¹³ = (20 × 16)¹³ = 320¹³

• Law 6 : When two different bases having equal exponents are in division, their division gets raised to the given exponent i.e., p^m/ q^m = (p/q)^m 

Example :

1. 12² / 30² = (12/30)² = 144/90
2.   5³ /2³  = (5/2)³ = 125/8

• Law 7: In case we are given a fractional exponent p^m/n =   .

Example:

1. 20^1/2 = √2
2. 20^1/3 = 
3. 692^44/5 = 

• Law 8 (Negative exponent law): Reciprocate the base if its exponent is negative, i.e., p^-m =     .

Example:

1. 2^-91 = 
2. 69420^-80 = 

Simplify 8/5x^-2/3

Solution:

Using the property a^-m = 1/ a^m, which is known as the Negative exponent law,

 8/ 5x^-2/3 = 

Therefore the answer is 

Solved Questions

Question 1: Simplify: 1/ 2x^-99.

Solution:

Using the property a^-m = 1/ a^m, which is known as the Negative exponent law,

1/ 2x^-99 = 

= x⁹⁹/ 2.

Question 2: Simplify: 4/3x^-9.

Solution:

Using the property a^-m = 1/ a^m, which is known as the Negative exponent law,

4/3x^-9 = 

Question 3: Simplify: 12x⁹/ 5x⁶⁰.

Solution:

Using the property a^m/ aⁿ = a^{m – n}, which is known as the quotient law,

12×9/ 5×60 = 

= 12x^-51/ 5

Using the property a^-m = 1/ a^m, which is known as the Negative exponent law,

12x^-51/ 5 = .

Question 4. Simplify: 3x²/ 10x⁵.

Solution:

Using the property a^m/ aⁿ = a^m-n, which is known as the quotient law,

3x²/ 10x⁵ = 

= 3x^-3/ 5

Using the property a^-m = 1/ a^m, which is known as the Negative exponent law,

3x^-3/ 5 = 

Question 5. Simplify: 2x⁴/ 5y^-10.

Solution:

Using the property a^-m = 1/ a^m, which is known as the Negative exponent law,

2x⁴/ 5y^-10 =