Question

Simplify and express the result in power notation with positive exponent. $ {( - 4)^5} \div {( - 4)^8} $

Answer 1

Hint: Here we are given two exponents and the division sign in between so first do the simplification using the law that when bases are the same and powers are subtracted in case of division.
The power is used to express mathematical equations in the short form; it is an expression that represents the repeated multiplication of the same factor. For example - $ 2 \times 2 \times 2 $ can be expressed as $ {2^3} $ . Here, the number two is called the base and the exponent represents the number of times the base is used as the factor.

Complete step by step solution:
Take the given expression: $ {( - 4)^5} \div {( - 4)^8} $
Re-write and frame the above given expression in the form of fraction: $ {( - 4)^5} \div {( - 4)^8} = \dfrac{{{{( - 4)}^5}}}{{{{( - 4)}^8}}} $
By using the law of the quotient of power rule: $ {( - 4)^5} \div {( - 4)^8} = {( - 4)^{5 - 8}} $
Simplify the above expression finding the subtraction for the power.
 $ {( - 4)^5} \div {( - 4)^8} = {( - 4)^{ - 3}} $
By using the law of the negative exponent rule which states that when the power and exponent moved to the denominator negative power becomes positive that is $ {a^{ - n}} = \dfrac{1}{{{a^n}}} $
 $ {( - 4)^5} \div {( - 4)^8} = \dfrac{1}{{{{( - 4)}^3}}} $
Simplify the above expression.
 $ {( - 4)^5} \div {( - 4)^8} = - \dfrac{1}{{64}} $
This is the required solution.
So, the correct answer is “ $ - \dfrac{1}{{64}} $ ”.

Note: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
I.Product of powers rule
II.Quotient of powers rule
III.Power of a power rule
IV.Power of a product rule
V.Power of a quotient rule
VI.Zero power rule
VII.Negative exponent rule