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

Question

Luna Angel · Accepted Answer

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

Here, we can make use of 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#

Zoe Adams · Answer

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

#= 512/19683# Here, we need to take into account four characteristics of indices, or exponents: 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#

Genesis Amaya · Answer

undefined To evaluate the expression $ \frac{(2/3)^4}{(2/3)^{-5}(2/3)^0} $ using the properties of indices, we can simplify the numerator and denominator separately and then divide the numerator by the denominator.

First, simplify the numerator:
$$ \left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} = \frac{16}{81} $$

Next, simplify the denominator:
$$ \left(\frac{2}{3}\right)^{-5} = \frac{1}{\left(\frac{2}{3}\right)^5} = \frac{1}{\frac{2^5}{3^5}} = \frac{3^5}{2^5} $$
$$ \left(\frac{2}{3}\right)^0 = 1 $$

Now, substitute the simplified values into the expression:
$$ \frac{\frac{16}{81}}{\frac{3^5}{2^5} \cdot 1} $$

Simplify the denominator:
$$ \frac{16}{81} \div \frac{3^5}{2^5} = \frac{16}{81} \times \frac{2^5}{3^5} = \frac{16 \times 2^5}{81 \times 3^5} $$

Calculate the powers of 2 and 3:
$$ 2^5 = 32 $$
$$ 3^5 = 243 $$

Substitute the values back into the expression:
$$ \frac{16 \times 32}{81 \times 243} $$

Calculate the numerator and denominator:
$$ 16 \times 32 = 512 $$
$$ 81 \times 243 = 19683 $$

Thus, the final result is:
$$ \frac{512}{19683} $$