Question

How do you simplify \[\dfrac{{5{x^{ - 4}}}}{{{x^{ - 9}}}}\] and write it using only positive exponents.

Answer 1

Hint: According to the question, we can see that it is in exponential form. To solve this question, we can use some laws of exponents. Here, we will use the law of exponential division to solve the question.
Formula used: \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]

Complete step by step solution:
The given exponent is \[\dfrac{{5{x^{ - 4}}}}{{{x^{ - 9}}}}\] . We are going to use the law of exponential division. The formula is:
 \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
This says that if the base number of both the numerator and denominator is the same, then the powers for the base will get subtracted, and we get the answer in the above given format.
Now, we will use the same formula in our question. We will put the values according to the formula, and we get:
 \[\dfrac{{5{x^{ - 4}}}}{{{x^{ - 9}}}}\]
Here to make the base same on both the numerator and the denominator, we will take 5 outside, and we get:
 \[ = 5 \times \dfrac{{{x^{ - 4}}}}{{{x^{ - 9}}}}\]
Now, we will solve this exponent according to the formula, and we get:
 \[ \Rightarrow 5 \times \dfrac{{{x^{ - 4}}}}{{{x^{ - 9}}}} = 5 \times {x^{ - 4 - ( - 9)}}\]
Now, we will add the powers in the right-hand side of the equation, and we get:
 \[ \Rightarrow 5 \times \dfrac{{{x^{ - 4}}}}{{{x^{ - 9}}}} = 5 \times {x^5}\]
We can also write this as:
 \[5 \times \dfrac{{{x^{ - 4}}}}{{{x^{ - 9}}}} = 5{x^5}\]
Therefore, we got our result that is \[\dfrac{{5{x^{ - 4}}}}{{{x^{ - 9}}}} = 5{x^5}\] .
So, the correct answer is “ \[\dfrac{{5{x^{ - 4}}}}{{{x^{ - 9}}}} = 5{x^5}\] ”.

Note: Exponential notation is a shorthand for very large and very small numbers. For small numbers, an example may be like the mass of an electron.
An exponent is any number that we can see in the top right corner of any other number which is called the base. In Mathematics, exponents can tell us how many times we can multiply the base with itself.