Join IDNLearn.com and start getting the answers you've been searching for. Explore a wide array of topics and find reliable answers from our experienced community members.
Sagot :
Sure, let's go through these step-by-step:
### a) [tex]\(\left(2^2\right)^3\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 2^2 = 4 \][/tex]
Next, raise that result to the power of 3:
[tex]\[ 4^3 = 64 \][/tex]
So, [tex]\(\left(2^2\right)^3 = 64 \)[/tex].
### b) [tex]\(\left(3^6\right)^{\frac{1}{2}}\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 3^6 = 729 \][/tex]
Next, take the square root of 729 (as [tex]\(\frac{1}{2}\)[/tex] represents the square root):
[tex]\[ \sqrt{729} = 27 \][/tex]
So, [tex]\(\left(3^6\right)^{\frac{1}{2}} = 27\)[/tex].
### c) [tex]\(\left(5^2\right)^{\frac{1}{2}}\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 5^2 = 25 \][/tex]
Next, take the square root of 25:
[tex]\[ \sqrt{25} = 5 \][/tex]
So, [tex]\(\left(5^2\right)^{\frac{1}{2}} = 5\)[/tex].
### d) [tex]\(\left(7^{-3}\right)^{\frac{1}{3}}\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 7^{-3} = \frac{1}{7^3} = \frac{1}{343} \][/tex]
Next, take the cube root of [tex]\(\frac{1}{343}\)[/tex] (as [tex]\(\frac{1}{3}\)[/tex] represents the cube root):
[tex]\[ \left(\frac{1}{343}\right)^{\frac{1}{3}} = \frac{1}{7} \approx 0.142857 \][/tex]
So, [tex]\(\left(7^{-3}\right)^{\frac{1}{3}} = 0.142857\)[/tex].
### f) [tex]\(4^{\frac{1}{2}}\)[/tex]
This expression represents the square root of 4:
[tex]\[ \sqrt{4} = 2 \][/tex]
So, [tex]\(4^{\frac{1}{2}} = 2\)[/tex].
### g) [tex]\(9^{\frac{1}{2}}\)[/tex]
This expression represents the square root of 9:
[tex]\[ \sqrt{9} = 3 \][/tex]
So, [tex]\(9^{\frac{1}{2}} = 3\)[/tex].
### h) [tex]\(8^{\frac{2}{3}}\)[/tex]
Express 8 as [tex]\(2^3\)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
Next, raise [tex]\(2^3\)[/tex] to the [tex]\(\frac{2}{3}\)[/tex] power:
[tex]\[ \left(2^3\right)^{\frac{2}{3}} = 2^{(3 \cdot \frac{2}{3})} = 2^2 = 4 \][/tex]
So, [tex]\(8^{\frac{2}{3}} = 4\)[/tex].
### i) [tex]\(16^{\frac{1}{4}}\)[/tex]
Express 16 as [tex]\(2^4\)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]
Next, take the fourth root of 16:
[tex]\[ \left(2^4\right)^{\frac{1}{4}} = 2^{(4 \cdot \frac{1}{4})} = 2^1 = 2 \][/tex]
So, [tex]\(16^{\frac{1}{4}} = 2\)[/tex].
### e) [tex]\(\left(\frac{2^2}{3^2}\right)^{-\frac{1}{2}}\)[/tex]
First, calculate the initial expression:
[tex]\[ \frac{2^2}{3^2} = \frac{4}{9} \][/tex]
Next, take the square root of [tex]\(\frac{4}{9}\)[/tex] and then take the reciprocal due to the negative exponent:
[tex]\[ \left(\frac{4}{9}\right)^{-\frac{1}{2}} = \frac{1}{\sqrt{\frac{4}{9}}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} = 1.5 \][/tex]
So, [tex]\(\left(\frac{2^2}{3^2}\right)^{-\frac{1}{2}} = 1.5\)[/tex].
### k) [tex]\(81^{-\frac{3}{4}}\)[/tex]
Express 81 as [tex]\(3^4\)[/tex]:
[tex]\[ 81 = 3^4 \][/tex]
Next, take the fourth root and then raise it to the negative third power:
[tex]\[ \left(3^4\right)^{-\frac{3}{4}} = 3^{(4 \cdot -\frac{3}{4})} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27} \approx 0.037037 \][/tex]
So, [tex]\(81^{-\frac{3}{4}} = 0.037037\)[/tex].
### l) [tex]\(\left(\frac{16}{25}\right)^{\frac{1}{2}}\)[/tex]
First, take the square root of [tex]\(\frac{16}{25}\)[/tex]:
[tex]\[ \left(\frac{16}{25}\right)^{\frac{1}{2}} = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} = 0.8 \][/tex]
So, [tex]\(\left(\frac{16}{25}\right)^{\frac{1}{2}} = 0.8\)[/tex].
### m) [tex]\(\left(\frac{16}{81}\right)^{-\frac{1}{4}}\)[/tex]
First, take the fourth root of [tex]\(\frac{16}{81}\)[/tex] and then take the reciprocal due to the negative exponent:
[tex]\[ \left(\frac{16}{81}\right)^{-\frac{1}{4}} = \frac{1}{\left(\frac{16}{81}\right)^{\frac{1}{4}}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \][/tex]
So, [tex]\(\left(\frac{16}{81}\right)^{-\frac{1}{4}} = 1.5\)[/tex].
### n) [tex]\(\left(\frac{625}{81}\right)^{-0.5}\)[/tex]
First, take the square root of [tex]\(\frac{625}{81}\)[/tex] and then take the reciprocal due to the negative exponent:
[tex]\[ \left(\frac{625}{81}\right)^{-0.5} = \frac{1}{\left(\frac{625}{81}\right)^{0.5}} = \frac{1}{\frac{25}{9}} = \frac{9}{25} = 0.36 \][/tex]
So, [tex]\(\left(\frac{625}{81}\right)^{-0.5} = 0.36\)[/tex].
### j) [tex]\(8^{-\frac{1}{3}}\)[/tex]
Express 8 as [tex]\(2^3\)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
Next, take the cube root and then take the reciprocal due to the negative exponent:
[tex]\[ \left(2^3\right)^{-\frac{1}{3}} = 2^{3 \cdot -\frac{1}{3}} = 2^{-1} = \frac{1}{2} \][/tex]
So, [tex]\(8^{-\frac{1}{3}} = 0.5\)[/tex].
### o) [tex]\(\left(\frac{243}{32}\right)^{-0.4}\)[/tex]
Take the expression to the power of [tex]\(-0.4\)[/tex]:
[tex]\[ \left(\frac{243}{32}\right)^{-0.4} = \left(\frac{32}{243}\right)^{0.4} \][/tex]
Although this result was directly computed, its numerical value is approximately:
[tex]\[ 0.444444 \][/tex]
So, [tex]\(\left(\frac{243}{32}\right)^{-0.4} = 0.444444\)[/tex].
### a) [tex]\(\left(2^2\right)^3\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 2^2 = 4 \][/tex]
Next, raise that result to the power of 3:
[tex]\[ 4^3 = 64 \][/tex]
So, [tex]\(\left(2^2\right)^3 = 64 \)[/tex].
### b) [tex]\(\left(3^6\right)^{\frac{1}{2}}\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 3^6 = 729 \][/tex]
Next, take the square root of 729 (as [tex]\(\frac{1}{2}\)[/tex] represents the square root):
[tex]\[ \sqrt{729} = 27 \][/tex]
So, [tex]\(\left(3^6\right)^{\frac{1}{2}} = 27\)[/tex].
### c) [tex]\(\left(5^2\right)^{\frac{1}{2}}\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 5^2 = 25 \][/tex]
Next, take the square root of 25:
[tex]\[ \sqrt{25} = 5 \][/tex]
So, [tex]\(\left(5^2\right)^{\frac{1}{2}} = 5\)[/tex].
### d) [tex]\(\left(7^{-3}\right)^{\frac{1}{3}}\)[/tex]
First, evaluate the exponent inside the parentheses:
[tex]\[ 7^{-3} = \frac{1}{7^3} = \frac{1}{343} \][/tex]
Next, take the cube root of [tex]\(\frac{1}{343}\)[/tex] (as [tex]\(\frac{1}{3}\)[/tex] represents the cube root):
[tex]\[ \left(\frac{1}{343}\right)^{\frac{1}{3}} = \frac{1}{7} \approx 0.142857 \][/tex]
So, [tex]\(\left(7^{-3}\right)^{\frac{1}{3}} = 0.142857\)[/tex].
### f) [tex]\(4^{\frac{1}{2}}\)[/tex]
This expression represents the square root of 4:
[tex]\[ \sqrt{4} = 2 \][/tex]
So, [tex]\(4^{\frac{1}{2}} = 2\)[/tex].
### g) [tex]\(9^{\frac{1}{2}}\)[/tex]
This expression represents the square root of 9:
[tex]\[ \sqrt{9} = 3 \][/tex]
So, [tex]\(9^{\frac{1}{2}} = 3\)[/tex].
### h) [tex]\(8^{\frac{2}{3}}\)[/tex]
Express 8 as [tex]\(2^3\)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
Next, raise [tex]\(2^3\)[/tex] to the [tex]\(\frac{2}{3}\)[/tex] power:
[tex]\[ \left(2^3\right)^{\frac{2}{3}} = 2^{(3 \cdot \frac{2}{3})} = 2^2 = 4 \][/tex]
So, [tex]\(8^{\frac{2}{3}} = 4\)[/tex].
### i) [tex]\(16^{\frac{1}{4}}\)[/tex]
Express 16 as [tex]\(2^4\)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]
Next, take the fourth root of 16:
[tex]\[ \left(2^4\right)^{\frac{1}{4}} = 2^{(4 \cdot \frac{1}{4})} = 2^1 = 2 \][/tex]
So, [tex]\(16^{\frac{1}{4}} = 2\)[/tex].
### e) [tex]\(\left(\frac{2^2}{3^2}\right)^{-\frac{1}{2}}\)[/tex]
First, calculate the initial expression:
[tex]\[ \frac{2^2}{3^2} = \frac{4}{9} \][/tex]
Next, take the square root of [tex]\(\frac{4}{9}\)[/tex] and then take the reciprocal due to the negative exponent:
[tex]\[ \left(\frac{4}{9}\right)^{-\frac{1}{2}} = \frac{1}{\sqrt{\frac{4}{9}}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} = 1.5 \][/tex]
So, [tex]\(\left(\frac{2^2}{3^2}\right)^{-\frac{1}{2}} = 1.5\)[/tex].
### k) [tex]\(81^{-\frac{3}{4}}\)[/tex]
Express 81 as [tex]\(3^4\)[/tex]:
[tex]\[ 81 = 3^4 \][/tex]
Next, take the fourth root and then raise it to the negative third power:
[tex]\[ \left(3^4\right)^{-\frac{3}{4}} = 3^{(4 \cdot -\frac{3}{4})} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27} \approx 0.037037 \][/tex]
So, [tex]\(81^{-\frac{3}{4}} = 0.037037\)[/tex].
### l) [tex]\(\left(\frac{16}{25}\right)^{\frac{1}{2}}\)[/tex]
First, take the square root of [tex]\(\frac{16}{25}\)[/tex]:
[tex]\[ \left(\frac{16}{25}\right)^{\frac{1}{2}} = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} = 0.8 \][/tex]
So, [tex]\(\left(\frac{16}{25}\right)^{\frac{1}{2}} = 0.8\)[/tex].
### m) [tex]\(\left(\frac{16}{81}\right)^{-\frac{1}{4}}\)[/tex]
First, take the fourth root of [tex]\(\frac{16}{81}\)[/tex] and then take the reciprocal due to the negative exponent:
[tex]\[ \left(\frac{16}{81}\right)^{-\frac{1}{4}} = \frac{1}{\left(\frac{16}{81}\right)^{\frac{1}{4}}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \][/tex]
So, [tex]\(\left(\frac{16}{81}\right)^{-\frac{1}{4}} = 1.5\)[/tex].
### n) [tex]\(\left(\frac{625}{81}\right)^{-0.5}\)[/tex]
First, take the square root of [tex]\(\frac{625}{81}\)[/tex] and then take the reciprocal due to the negative exponent:
[tex]\[ \left(\frac{625}{81}\right)^{-0.5} = \frac{1}{\left(\frac{625}{81}\right)^{0.5}} = \frac{1}{\frac{25}{9}} = \frac{9}{25} = 0.36 \][/tex]
So, [tex]\(\left(\frac{625}{81}\right)^{-0.5} = 0.36\)[/tex].
### j) [tex]\(8^{-\frac{1}{3}}\)[/tex]
Express 8 as [tex]\(2^3\)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
Next, take the cube root and then take the reciprocal due to the negative exponent:
[tex]\[ \left(2^3\right)^{-\frac{1}{3}} = 2^{3 \cdot -\frac{1}{3}} = 2^{-1} = \frac{1}{2} \][/tex]
So, [tex]\(8^{-\frac{1}{3}} = 0.5\)[/tex].
### o) [tex]\(\left(\frac{243}{32}\right)^{-0.4}\)[/tex]
Take the expression to the power of [tex]\(-0.4\)[/tex]:
[tex]\[ \left(\frac{243}{32}\right)^{-0.4} = \left(\frac{32}{243}\right)^{0.4} \][/tex]
Although this result was directly computed, its numerical value is approximately:
[tex]\[ 0.444444 \][/tex]
So, [tex]\(\left(\frac{243}{32}\right)^{-0.4} = 0.444444\)[/tex].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.
