Question
Evaluate the left hand side to find the value of a,a in the equation in simplest form.
start fraction, x, start superscript, start fraction, 5, divided by, 3, end fraction, end superscript, divided by, x, start superscript, start fraction, 6, divided by, 5, end fraction, end superscript, end fraction, equals, x, to the power a
x
5
6
x
3
5
=
x
a
start fraction, x, start superscript, start fraction, 5, divided by, 3, end fraction, end superscript, divided by, x, start superscript, start fraction, 6, divided by, 5, end fraction, end superscript, end fraction, equals, x, to the power a
x
5
6
x
3
5
=
x
a
Answers
GPT-4o mini
We start with the equation:
\[
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^a
\]
To simplify the left-hand side, we use the property of exponents that states \(\frac{x^m}{x^n} = x^{m-n}\). Accordingly, we can write:
\[
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^{\frac{5}{3} - \frac{6}{5}}
\]
Next, we need to find a common denominator to subtract the exponents. The denominators are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15.
Now we can rewrite each fraction:
\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\]
\[
\frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15}
\]
Now we can perform the subtraction:
\[
\frac{5}{3} - \frac{6}{5} = \frac{25}{15} - \frac{18}{15} = \frac{25 - 18}{15} = \frac{7}{15}
\]
Thus, we have:
\[
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^{\frac{7}{15}}
\]
Now we can equate this to the right-hand side of the original equation:
\[
x^{\frac{7}{15}} = x^a
\]
Since the bases \(x\) are the same, we can equate the exponents:
\[
\frac{7}{15} = a
\]
Therefore, the final value of \(a\) in simplest form is:
\[
\boxed{\frac{7}{15}}
\]
\[
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^a
\]
To simplify the left-hand side, we use the property of exponents that states \(\frac{x^m}{x^n} = x^{m-n}\). Accordingly, we can write:
\[
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^{\frac{5}{3} - \frac{6}{5}}
\]
Next, we need to find a common denominator to subtract the exponents. The denominators are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15.
Now we can rewrite each fraction:
\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\]
\[
\frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15}
\]
Now we can perform the subtraction:
\[
\frac{5}{3} - \frac{6}{5} = \frac{25}{15} - \frac{18}{15} = \frac{25 - 18}{15} = \frac{7}{15}
\]
Thus, we have:
\[
\frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}} = x^{\frac{7}{15}}
\]
Now we can equate this to the right-hand side of the original equation:
\[
x^{\frac{7}{15}} = x^a
\]
Since the bases \(x\) are the same, we can equate the exponents:
\[
\frac{7}{15} = a
\]
Therefore, the final value of \(a\) in simplest form is:
\[
\boxed{\frac{7}{15}}
\]