Find answers to your questions faster and easier with IDNLearn.com. Join our Q&A platform to access reliable and detailed answers from experts in various fields.

Let's evaluate the following expressions:

a) [tex]\left(2^2\right)^3[/tex]

b) [tex]\left(3^6\right)^{\frac{1}{2}}[/tex]

c) [tex]\left(5^2\right)^{\frac{1}{2}}[/tex]

d) [tex]\left(7^{-3}\right)^{\frac{1}{3}}[/tex]

e) [tex]\left(\frac{2^2}{3^2}\right)^{-\frac{1}{2}}[/tex]

f) [tex]4^{\frac{1}{2}}[/tex]

g) [tex]9^{\frac{1}{2}}[/tex]

h) [tex]8^{\frac{2}{3}}[/tex]

i) [tex]16^{\frac{1}{4}}[/tex]

j) [tex]8^{-\frac{1}{3}}[/tex]

k) [tex]81^{-\frac{3}{4}}[/tex]

l) [tex]\left(\frac{16}{25}\right)^{\frac{1}{2}}[/tex]

m) [tex]\left(\frac{16}{81}\right)^{-\frac{1}{4}}[/tex]

n) [tex]\left(\frac{625}{81}\right)^{-0.5}[/tex]

o) [tex]\left(\frac{243}{32}\right)^{-0.4}[/tex]


Sagot :

Sure, let's solve the given expressions step-by-step for each part.

a) [tex]\(\left(2^2\right)^3\)[/tex]

First, calculate [tex]\(2^2\)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]

Then, raise this result to the power 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, calculate [tex]\(3^6\)[/tex]:
[tex]\[ 3^6 = 729 \][/tex]

Then, take 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, calculate [tex]\(5^2\)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]

Then, take the square root:
[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, calculate [tex]\(7^{-3}\)[/tex]:
[tex]\[ 7^{-3} = \frac{1}{7^3} = \frac{1}{343} \][/tex]

Then, take the cube root:
[tex]\[ \sqrt[3]{\frac{1}{343}} = \frac{1}{\sqrt[3]{343}} = \frac{1}{7} \][/tex]

So, [tex]\(\left(7^{-3}\right)^{\frac{1}{3}} \approx 0.14285714285714288\)[/tex].

f) [tex]\(4^{\frac{1}{2}}\)[/tex]

Take 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]

Take 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]

First, express 8 as a power of 2:
[tex]\[ 8 = 2^3 \][/tex]

Then, [tex]\(8^{\frac{2}{3}}\)[/tex] becomes:
[tex]\[ (2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4 \][/tex]

So, [tex]\(8^{\frac{2}{3}} \approx 3.9999999999999996\)[/tex].

i) [tex]\(16^{\frac{1}{4}}\)[/tex]

First, calculate [tex]\(16\)[/tex] as a power of [tex]\(2\)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]

Then, take the 4th root:
[tex]\[ \sqrt[4]{16} = \sqrt[4]{2^4} = 2 \][/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 fraction inside:
[tex]\[ \frac{2^2}{3^2} = \frac{4}{9} \][/tex]

Now, raise to the power [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[ \left(\frac{4}{9}\right)^{-\frac{1}{2}} = \left(\frac{9}{4}\right)^{\frac{1}{2}} = \sqrt{\frac{9}{4}} = \frac{3}{2} \][/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 a power:
[tex]\[ 81 = 3^4 \][/tex]

Then, calculate:
[tex]\[ 81^{-\frac{3}{4}} = (3^4)^{-\frac{3}{4}} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27} \][/tex]

So, [tex]\(81^{-\frac{3}{4}} \approx 0.037037037037037035\)[/tex].

l) [tex]\(\left(\frac{16}{25}\right)^{\frac{1}{2}}\)[/tex]

Take the square root of the fraction:
[tex]\[ \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \][/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]

Take the reciprocal of the fraction and then the 4th root:
[tex]\[ \left(\frac{16}{81}\right)^{-\frac{1}{4}} = \left(\frac{81}{16}\right)^{\frac{1}{4}} = \frac{\sqrt[4]{81}}{\sqrt[4]{16}} = \frac{3}{2} \][/tex]

So, [tex]\(\left(\frac{16}{81}\right)^{-\frac{1}{4}} = 1.5\)[/tex].

n) [tex]\(\left(\frac{625}{81}\right)^{-0.5}\)[/tex]

Take the reciprocal of the fraction and then the square root:
[tex]\[ \left(\frac{625}{81}\right)^{-0.5} = \left(\frac{81}{625}\right)^{0.5} = \sqrt{\frac{81}{625}} = \frac{9}{25} \][/tex]

So, [tex]\(\left(\frac{625}{81}\right)^{-0.5} = 0.36\)[/tex].

j) [tex]\(8^{-\frac{1}{3}}\)[/tex]

Express 8 as a power:
[tex]\[ 8 = 2^3 \][/tex]

Then, calculate the cube root and apply the negative exponent:
[tex]\[ 8^{-\frac{1}{3}} = (2^3)^{-\frac{1}{3}} = 2^{-1} = \frac{1}{2} \][/tex]

So, [tex]\(8^{-\frac{1}{3}} = 0.5\)[/tex].

o) [tex]\(\left(\frac{243}{32}\right)^{-0.4}\)[/tex]

First, raise the fraction to the power [tex]\(-0.4\)[/tex]:
[tex]\[ \left(\frac{243}{32}\right)^{-0.4} \approx 0.4444444444444444 \][/tex]

So, [tex]\(\left(\frac{243}{32}\right)^{-0.4} \approx 0.4444444444444444\)[/tex].