Question

simplify using only POSITIVE exponents:

1. (2x^3y^7)^-2

2. 12x^5y^3
--------------
4x^-1

3. (r^-7 B^-8)
---------------
(t^-4 w)

Answer 1

So here are a few rules with exponents that you should know:

1. Multiplying exponents of the same base: $x^m*x^n=x^{m+n}$
2. Dividing exponents of the same base: $x^m\div x^n=x^{m-n}$
3. Powering a power to a power: $(x^m)^n=x^{m*n}$
4. Converting a negative exponent to a positive one: $x^{-m}=\frac{1}{x^m}\ \textsf{and}\ \frac{1}{x^{-m}}=x^m$

1.

Firstly, solve the outside exponent:

$(2x^3y^7)^{-2}=2^{-2}x^{3*-2}y^{7*-2}=2^{-2}x^{-6}y^{-14}$

Next, convert the negative exponents into positive ones:

$2^{-2}x^{-6}y^{-14}=\frac{1}{2^2x^6y^{14}}=\frac{1}{4x^6y^{14}}$

Your final answer is $\frac{1}{4x^6y^{14}}$

2.

For this, just divide:

$\frac{12x^5y^3}{4x^{-1}}=\frac{12}{4}x^{5-(-1)}y^{3-0}=3x^6y^3$

Your final answer is $3x^6y^3$

3.

For this, convert all negative exponents into positive ones:

$\frac{r^{-7}b^{-8}}{t^{-4}w}=\frac{t^4}{r^7b^8w}$

Your final answer is $\frac{t^4}{r^7b^8w}$