Question
Question
(1 point)
Evaluate the left-hand side to find the value of a in the equation in simplest form
x5/3/x6/5 =x^a
a=4/15
A is equal to 4 over 15
a=43/15
A is equal to 43 over 15
a=7/15
A is equal to 7 over 15
a=1/2
Answers
GPT-4o mini
To solve the equation \(\frac{x^{5/3}}{x^{6/5}} = x^a\), we can use the property of exponents \(\frac{x^m}{x^n} = x^{m-n}\).
Now, we can rewrite the left-hand side:
\[
\frac{x^{5/3}}{x^{6/5}} = x^{5/3 - 6/5}
\]
Next, we need to compute \(5/3 - 6/5\). To perform this subtraction, we need a common denominator. The least common multiple of 3 and 5 is 15. So we will convert each fraction:
\[
5/3 = \frac{5 \cdot 5}{3 \cdot 5} = \frac{25}{15}
\]
\[
6/5 = \frac{6 \cdot 3}{5 \cdot 3} = \frac{18}{15}
\]
Now we can subtract:
\[
\frac{25}{15} - \frac{18}{15} = \frac{25 - 18}{15} = \frac{7}{15}
\]
So, we have:
\[
\frac{x^{5/3}}{x^{6/5}} = x^{\frac{7}{15}}
\]
Therefore, the equation simplifies to:
\[
x^{\frac{7}{15}} = x^a
\]
This implies that:
\[
a = \frac{7}{15}
\]
Thus, the value of \(a\) is \(\frac{7}{15}\).
So the correct answer is:
\(a = 7/15\)
**A is equal to 7 over 15.**
Now, we can rewrite the left-hand side:
\[
\frac{x^{5/3}}{x^{6/5}} = x^{5/3 - 6/5}
\]
Next, we need to compute \(5/3 - 6/5\). To perform this subtraction, we need a common denominator. The least common multiple of 3 and 5 is 15. So we will convert each fraction:
\[
5/3 = \frac{5 \cdot 5}{3 \cdot 5} = \frac{25}{15}
\]
\[
6/5 = \frac{6 \cdot 3}{5 \cdot 3} = \frac{18}{15}
\]
Now we can subtract:
\[
\frac{25}{15} - \frac{18}{15} = \frac{25 - 18}{15} = \frac{7}{15}
\]
So, we have:
\[
\frac{x^{5/3}}{x^{6/5}} = x^{\frac{7}{15}}
\]
Therefore, the equation simplifies to:
\[
x^{\frac{7}{15}} = x^a
\]
This implies that:
\[
a = \frac{7}{15}
\]
Thus, the value of \(a\) is \(\frac{7}{15}\).
So the correct answer is:
\(a = 7/15\)
**A is equal to 7 over 15.**