How do you evaluate the expression $(2/3)^4/((2/3)^-5(2/3)^0)$ using the properties of indices?.

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Answer 1 · 2017-05-23T14:32:23

$(2/3)^4/((2/3)^-5(2/3)^0)=(2/3)^9=512/19683$

We can use here the identities

$a^mxxa^n=a^((m+n))$ and $a^m/a^n=a^((m-n))$

As such $a^m/(a^na^p)=a^((m-n-p))$

Hence $(2/3)^4/((2/3)^-5(2/3)^0)$

$=(2/3)^((4-(-5)-0))$

$=(2/3)^((4+5))$

$=(2/3)^9$

or $2^9/3^9=512/19683$

Answer 2 · 2017-05-23T14:38:44

$2^9/3^9 = (2/3)^9$

$= 512/19683$

There are four properties of indices (exponents) to consider here:

Raising factors to a power: $color(blue)((xy)^m = x^m xx y^m)$

A negative index: $color(magenta)(1/x^-m = x^m)$

Index of $0$ . $color(lime)("Anything to power of 0 is equal to"1)" "$ (except $0^0$ )

Multiply law: same bases, add the indices: $x^m xx x^n = x^(m+n)$

$color(blue)((2/3)^4)/(color(magenta)((2/3)^-5)color(lime)((2/3)^0)) = (color(blue)(2^4/3^4)xxcolor(magenta)((2/3)^5))/(color(lime)((1))$

$=2^4/3^4xx2^5/3^5$

$= 2^9/3^9$

This can also be written as $(2/3)^9$

$= 512/19683$

How do you evaluate the expression (2/3)^4/((2/3)^-5(2/3)^0)(2/3)^4/((2/3)^-5(2/3)^0) using the properties of indices?.