Express with only positive exponents. (32 x4 y4 )-3 / (33 x-5 y5)-5

Question

(3² x⁴ y⁴ )^-3 / (3³ x-⁵ y⁵)^-5

Answer 1

= 3^9(y^13)/x^37= 19,683y^13/x^37

for

(a^c)^b = a^cb exponents multiply

a^d times a^e add the exponents = a^(d+e)

a^d/a^e subtract exponents = a^(d-e)

Answer 2

Answer 3

Let's just grab the bull by the horns and tackle this!

Let's deal with the first piece.

We remember that to raise a power to a power, we multiply the exponents, so the numerator is:

(3²x⁴y⁴)^-3

= 3^2(-3) x^{4(-3 )}y^4(-3)

= 3^-6 x^-12 y^-12

The bottom piece: well, let's flip it around to the numerator, by changing the sign of the exponent:

1 / (3³ x^-5 y⁵)^-5 = (3³ x^-5 y⁵)⁵

= 3³⁽⁵⁾ x^(-5)5 y⁵⁽⁵⁾

= 3¹⁵ x^-25 y²⁵

Multiply it out (add the exponents of like terms):

(3^-6 x^-12 y^-12 ) ( 3¹⁵ x^-25 y²⁵ )

= 3^-6+15 x^-12-25 y^-12+25

= 3⁹ x^-37 y¹³

We could stop there, but we were asked to use only positive exponents.

This means negative exponents get put back in the denominator, with their signs changed:

So 3⁹ x^-37 y¹³

.........3⁹ y¹³

= ----------------

..........x³⁷

Answer 4

I think the video shows the wrong exponent on the power in the denominator (should be 37)

Here is a Desmos graph showing different flavors for this algebraic expression:

desmos.com/calculator/ehnx2hf657