Questions: The expression [ left(frac2 a^-65 b^-1 / 2right)^-1 ] equals (n a^r / b^t) where n , the coefficient, is: (r), the exponent of (a), is: (t), the exponent of (b), is:

$The expression [ left(frac2 a^-65 b^-1 / 2right)^-1 ] equals (n a^r / b^t) where n , the coefficient, is: (r), the exponent of (a), is: (t), the exponent of (b), is:$

Transcript text: The expression \[ \left(\frac{2 a^{-6}}{5 b^{-1 / 2}}\right)^{-1} \] equals $n a^{r} / b^{t}$ where n , the coefficient, is: $\square$ $r$, the exponent of $a$, is: $\square$ $t$, the exponent of $b$, is: $\square$

Solution

Solution Steps

Step 1: Rewrite the expression using the negative exponent rule

The given expression is: \[ \left(\frac{2 a^{-6}}{5 b^{-1 / 2}}\right)^{-1} \] Using the rule $ \left(\frac{x}{y}\right)^{-1} = \frac{y}{x} $, we rewrite the expression as: \[ \frac{5 b^{-1 / 2}}{2 a^{-6}} \]

Step 2: Simplify the exponents of $ a $ and $ b $

Using the rule $ x^{-n} = \frac{1}{x^n} $, we rewrite the expression as: \[ \frac{5 \cdot \frac{1}{b^{1/2}}}{2 \cdot \frac{1}{a^6}} \] This simplifies to: \[ \frac{5}{2} \cdot \frac{a^6}{b^{1/2}} \]

Step 3: Express the result in the form $ \frac{n a^r}{b^t} $

The expression is now in the form: \[ \frac{5}{2} \cdot \frac{a^6}{b^{1/2}} = \frac{5 a^6}{2 b^{1/2}} \] Thus: