2 questions Exponents

Question

Express using positive exponents. Then if possible simplify.
5a^-7 b^4

Odd and even Nth root
4 (7b)^4

Answer 1

x^-n means 1⁄xⁿ. Verbally, this works out to mean that "in context of a fraction, we can change the sign of the exponent, by moving that factor (with its base) to the other side of the fraction bar where sign of the exponent will change (the base stays the same)." Soince 5a^-7b⁴ is (5a^-7b⁴)/1. We can apply that here with the -7 power.

 

So (5a^-7b⁴)/1 is equivalent to (5b⁴)/(a⁷).

 

Note that the exponent of -7 changed to positive 7 because we "moved" the power of "a" to the other side of the fraction bar. Also notice that the 5 is still in the numerator, since 5 was never part of the base of the -7 power.

 

 

It is not clear what the "Odd and even Nth root" comment relates to, since there are no roots in either problem.

 

For the second problem, it is helpful to remember that (xy)ⁿ=xⁿyⁿ (verbally "exponents distribute over multiplication). This makes 4(7b)⁴=4*7⁴b⁴. Since 4*7⁴=9604, 4(7b)⁴=9604b⁴.