QuestionDance 5th Period
Evaluate the left hand side to find the value of $a$ in the equation in simplest forn $\frac{x^{\frac{3}{2}}}{x^{\frac{2}{5}}}=x^{a}$

Studdy Solution

STEP 1

1. The equation involves simplifying an expression with exponents.
2. The properties of exponents will be used to simplify the expression.

STEP 2

1. Apply the quotient rule for exponents.
2. Simplify the expression to find the value of $a$ .

STEP 3

Apply the quotient rule for exponents, which states that $\frac{x^m}{x^n} = x^{m-n}$ . In this case, $m = \frac{3}{2}$ and $n = \frac{2}{5}$ :
$\frac{x^{\frac{3}{2}}}{x^{\frac{2}{5}}} = x^{\left(\frac{3}{2} - \frac{2}{5}\right)}$

STEP 4

To simplify $\frac{3}{2} - \frac{2}{5}$ , find a common denominator. The least common denominator of 2 and 5 is 10:
$\frac{3}{2} = \frac{15}{10}, \quad \frac{2}{5} = \frac{4}{10}$
Now subtract the fractions:
$\frac{3}{2} - \frac{2}{5} = \frac{15}{10} - \frac{4}{10} = \frac{11}{10}$

STEP 5

Substitute the simplified exponent back into the expression:
$x^{\left(\frac{3}{2} - \frac{2}{5}\right)} = x^{\frac{11}{10}}$
Thus, the value of $a$ is:
$a = \frac{11}{10}$
The value of $a$ is:
$\boxed{\frac{11}{10}}$

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