Certainly! Let's express [tex]\(\sqrt{8}\)[/tex] as a power of 2 through a step-by-step process:
First, we start by expressing 8 in terms of powers of 2. We know that:
[tex]\[ 8 = 2^3 \][/tex]
Now, we need to find the square root of 8. So we write:
[tex]\[ \sqrt{8} = \sqrt{2^3} \][/tex]
Next, we use the property of exponents that states [tex]\(\sqrt{a} = a^{1/2}\)[/tex]. Applying this to our expression, we get:
[tex]\[ \sqrt{2^3} = (2^3)^{1/2} \][/tex]
When raising a power to another power, we multiply the exponents. Therefore:
[tex]\[ (2^3)^{1/2} = 2^{3 \cdot \frac{1}{2}} \][/tex]
Simplifying the multiplication in the exponent, we obtain:
[tex]\[ 2^{3/2} \][/tex]
So, [tex]\(\sqrt{8}\)[/tex] expressed as a power of 2 is:
[tex]\[ \sqrt{8} = 2^{3/2} \][/tex]
If you calculate [tex]\(2^{3/2}\)[/tex], you will get approximately:
[tex]\[ 2.8284271247461903 \][/tex]
Thus, [tex]\(\sqrt{8}\)[/tex] as a power of 2 is exactly [tex]\(2^{3/2}\)[/tex].
