Question

Simplify (15x^-4)(x^15)/(5x^4)(x^5)

Answer 1

`3x^2`

Explanation:

First main thing to know is the product and quotient rule of exponents.

Product Rule:

`x^a*x^b = x^(a+b)`

And if this doesn't make sense, you can think of the exponent like this:

`x^a*x^b = (x*x*x*x...\text{ a amount of times}) * (x * x * x \text{ b amount of times})`

and since multiplication is commutative, we can just combine all these x's, and since the total amount on the left is "a", and the right is "b", the total combined x's should be a+b, which can be expressed as:

`x*x*x... \text{ a+b amount of times}`

which can be expressed as an exponent (x^(a+b))

Quotient Rule:

`(x^a)/(x^b) = x^(a-b)`

You can use similar reasoning for this, since if you write it out you get

`\frac{x*x*x...\text{ a amount of times}}{x*x*x\text{ b amount of times}}`

and since you have an x in the numerator and the denominator, you can simply cancel the x's out. In doing this you want to remove the denominator, so you cancel out "b" x's. So there will be (a-b) x's left in the numerator, and a 1 in the denominator, so it's just x^(a-b)

Ok so now let's apply these to solve your question

`((15x^(-4))*x^(15))/((5x^4)*x^5)\\`

So let's combine the exponents in the numerator and denominator using the product rule

`(15x^(11))/(5x^9)\\`

Now we can divide the 15 by 5, and divide the x^11 by the x^9 using the quotient rule

`3x^2`