Certainly! Let's solve the problem step by step:
Given equation:
[tex]$
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^a
$[/tex]
To find the value of [tex]\( a \)[/tex], we will use the properties of exponents. When you divide powers with the same base, you subtract the exponents. So we have:
[tex]$
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^{\frac{5}{3} - \frac{6}{5}}
$[/tex]
Next, we need to subtract the exponents. To do this, we should get a common denominator for the fractions [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{6}{5}\)[/tex].
The least common denominator of 3 and 5 is 15. Let's convert both fractions to have this common denominator:
[tex]$
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
$[/tex]
[tex]$
\frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15}
$[/tex]
Now, we can subtract the fractions:
[tex]$
\frac{25}{15} - \frac{18}{15} = \frac{25 - 18}{15} = \frac{7}{15}
$[/tex]
So, we can simplify the original expression as follows:
[tex]$
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^{\frac{7}{15}}
$[/tex]
Thus, comparing exponents on both sides of the original equation, we find that:
[tex]$
a = \frac{7}{15}
$[/tex]
Converting [tex]\(\frac{7}{15}\)[/tex] to a decimal:
[tex]$
a = 0.4666666666666667
$[/tex]
Therefore, the value of [tex]\( a \)[/tex] is:
[tex]$
a = 0.4666666666666667
$[/tex]
