How do you simplify #((3^6)^n times (81)^(2n)) / (3^n)^4#?

Answer 1

Eli Assouline

# = color(green)(3^(10n)#

#((3^6)^n xx (81)^(2n)) /((3^n)^4#

#81 = 3 * 3 * 3 * 3 = color(blue)(3^4#

So, #81 ^(2n) = (3^4 )^(2n)#

# (3^4 )^(2n) = color(green)( 3 ^(8n)#

#(3^6)^n = 3^(6n)#

#(3^n)^4 = 3^(4n)#

The expression can now be written as:

#((3^6)^n xx (81)^(2n)) /((3^n)^4 ) = (3^(6n) xx color(green)( 3 ^(8n))) /(3^ ( 4n)#

# =3^((6n + 8n)) /(3^ ( 4n)#

# =3^(14n) /(3^ ( 4n)#

# =3^((14n - 4n))#

# = color(green)(3^(10n)#

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Answer 2

Ethan Adkins

To simplify the expression, we apply the properties of exponents:

((3^6)^n * (81)^(2n)) / (3^n)^4

= (3^(6n) * 3^(8n)) / 3^(4n)

= 3^(6n + 8n - 4n)

= 3^(10n)

So, the simplified expression is 3^(10n).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.