Evaluate the left hand side to find the value of a,a in the equation in simplest form.

Question

start fraction, x, start superscript, start fraction, 5, divided by, 3, end fraction, end superscript, divided by, x, start superscript, start fraction, 3, divided by, 2, end fraction, end superscript, end fraction, equals, x, to the power a
x
2
3
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x
3
5
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=
x
a

Answer 1

To evaluate the left-hand side of the equation given:

\[ \frac{x^{\frac{5}{3}}}{x^{\frac{3}{2}}} \]

we use the property of exponents that states:

\[ \frac{x^m}{x^n} = x^{m-n} \]

In this case, let \( m = \frac{5}{3} \) and \( n = \frac{3}{2} \). We first need to find \( m - n \):

1. Convert both fractions to have a common denominator. The least common multiple (LCM) of 3 and 2 is 6.

• For \( \frac{5}{3} \), convert it: \[ \frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6} \]

• For \( \frac{3}{2} \), convert it: \[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]

2. Now subtract the two fractions: \[ \frac{10}{6} - \frac{9}{6} = \frac{10 - 9}{6} = \frac{1}{6} \]

Thus, we have: \[ \frac{x^{\frac{5}{3}}}{x^{\frac{3}{2}}} = x^{\frac{1}{6}} \]

So now we can write our original equation as: \[ x^{\frac{1}{6}} = x^a \]

Since the bases are the same, we equate the exponents: \[ \frac{1}{6} = a \]

Therefore, the value of \( a \) is: \[ \boxed{\frac{1}{6}} \]