Can you help me solve this . Please?

Question

- (8a^3b^6)^ -2/3

*Simplify each expression. All variables represent positive real numbers. *

Answer 1

Rules here are:

(xy)^a = (x^a) * (y^a)

x^(-a) = 1/(x^a)

and

(a^x)^y = a^(xy)

So first, let's get rid of that negative exponent:

- (8a^3b^6)^(-2/3)

-1 / [(8a^3b^6)^(2/3)]

Now, I'll change the 8 into 2^3, this will make sense shortly:

-1 / [(2^3a^3b^6)^(2/3)]

Now, let's move the 2/3 exponent into the parenthesis:

-1 / [(2^(3 * 2/3) a^(3 * 2/3) b^(6 * 2/3)]

simplify:

-1 / [2^2 a^2 b^4]

-1 / (4 a^2 b^4)

Answer 2

If you raise the product to a certain power, that is the same as raising every multiplier to that power and then multiply them, i.e. (ab)^x=a^x*b^x

Another rule of exponents is that (a^x)^y = a^(xy), i.e. a variable raised to a power and then raised again is the same as raising the variable to a power, equal to the product of the powers.

Also, raising a number to a negative power means raising the reciprocal number to the opposite of the power, i.e. a^(-x)=(1/a)^x

Raising a number to a power that is a fraction means that we calculate the number raised to the power equal to the numerator and then getting a root equal to the denominator of the result, i.e. a^(x/y) = y root of (a^x)

So, here we can use these rules to get:

- (8a^3b^6)^ -2/3 = - [8^(-2/3)*(a^3)^(-2/3)*(b^6)^(-2/3)] =

= - [(1/8)^(2/3)*a^(3*(-2/3))*b^(6*(-2/3))] =

= - [third root of (1/8^2)*a^(-2)*b^(-4)]=

= - [third root of 1/64 *a^(-2)*b^(-4)] = -1/4*a^(-2)*b^(-4)=

= -1/4*1/a^2*1/b^4

You are told that a>o and b>0 because you cannot raise a negative variable to a power that is a fraction.

Hope that helps you :)