Question

How do you write $ {( - 2x)^{ - 4}} $ with positive exponents?

Answer 1

Hint: Here we will use 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}}} $ and then simplify the expression for the resultant required value.

Complete step-by-step solution:
Take the given expression: $ {( - 2x)^{ - 4}} $
Using the law of negative exponent rule,
 $ \Rightarrow {( - 2x)^{ - 4}} = \dfrac{1}{{{{( - 2x)}^4}}} $
Simplify the above expression, remember when there is power outside the whole bracket then power is applied to both the terms variable and the constant in the above expression.
 $ \Rightarrow {( - 2x)^{ - 4}} = \dfrac{1}{{{{( - 2)}^4}{{(x)}^4}}} $
When any negative term is multiplied even times, then the resultant value will be positive since the product of negative term with the negative term gives the positive term. Simply, the product of minus with minus gives plus.
\[{( - 2x)^{ - 4}} = \dfrac{1}{{16{x^4}}}\]
This is the required solution.

Note: 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. 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.
-Product of powers rule
-Quotient of powers rule
-Power of a power rule
-Power of a product rule
-Power of a quotient rule
-Zero power rule
-Negative exponent rule