Answer:
[tex](8^{\frac{2}{3} } )^{4} = 256[/tex]
Step-by-step explanation:
Given
[tex](8^{\frac{2}{3} } )^{4}[/tex]
Required
Simplify
To simplify this, we apply law of indices but first we start by solving the expression in bracket.
8 =2 * 2 * 2
8 = 2³
So, we substitute 2³ for 8
[tex](8^{\frac{2}{3} } )^{4}[/tex] becomes
[tex]((2^{3})^{\frac{2}{3} } )^{4}[/tex]
From law of indices
[tex]a^{n} = (a^{m})^{\frac{n}{m} }[/tex] ==> [tex](a^{m})^{\frac{n}{m} } = a^{n}[/tex]
So, [tex](2^{3})^{\frac{2}{3} } = 2^{2}[/tex]
At this point we have
[tex]((2^{3})^{\frac{2}{3} } )^{4}[/tex] = [tex](2^{2})^{4}[/tex]
Also, from law of indices
[tex](a^{m})^{n} = a^{m.n}[/tex]
So,
[tex]((2^{3})^{\frac{2}{3} } )^{4}[/tex] = [tex](2^{2})^{4}[/tex]
[tex](2^{2})^{4} = 2^{2*4}[/tex]
[tex](2^{2})^{4} = 2^{8}[/tex]
[tex](2^{2})^{4} = 256[/tex]
Hence,
[tex](8^{\frac{2}{3} } )^{4} = 256[/tex]
