Question

How do you simplify `\frac { q \cdot q ^ { 54} } { q \cdot q ^ { - 7} }`?

Answer 1

See some solution processes below:

Explanation:

First, use this rule of exponents to rewrite the expression:

`a = a^color(red)(1)`

`(q^color(red)(1) * q^color(blue)(54))/(q^color(red)(1) * q^color(blue)(-7))`

Next, use this rule of exponents to simplify the numerator and denominator:

`x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`(q^color(red)(1) * q^color(blue)(54))/(q^color(red)(1) * q^color(blue)(-7)) => q^(color(red)(1) + color(blue)(54))/q^(color(red)(1) + color(blue)(-7)) => q^color(red)(55)/q^color(blue)(-6)`

Now, use this rule of exponents to simplify the expression:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))`

`q^color(red)(55)/q^color(blue)(-6) => q^(color(red)(55)-color(blue)(-6)) => q^(color(red)(55)+color(blue)(6)) => q^61`

Another second process would be to first cancel common terms in the numerator and denominator:

`(color(red)(cancel(color(black)(q))) * q^color(red)(54))/(color(red)(cancel(color(black)(q))) * q^color(blue)(-7)) => q^color(red)(54)/q^color(blue)(-7)`

Then use this rule of exponents to complete the simplification:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))`

`q^color(red)(54)/q^color(blue)(-7) => q^(color(red)(54)-color(blue)(-7)) => q^(color(red)(54)+color(blue)(7)) => q^61`