b) $$ \left(\frac{-3}{5} ight)^{14} imes\left(\frac{-3}{5} ight)^{-3}=\left(\frac{-3}{5} ight)^{3 m+2} $$

Question

b) $$ \left(\frac{-3}{5}ight)^{14} 	imes\left(\frac{-3}{5}ight)^{-3}=\left(\frac{-3}{5}ight)^{3 m+2} $$

UpStudy AI · Accepted Answer

We start with the expression:
$$
\left(\frac{-3}{5}\right)^{14} \times \left(\frac{-3}{5}\right)^{-3} = \left(\frac{-3}{5}\right)^{3 m+2}
$$

Recall that when multiplying exponentials with the same base, we add the exponents. Thus, the left-hand side simplifies as follows:
$$
\left(\frac{-3}{5}\right)^{14} \times \left(\frac{-3}{5}\right)^{-3} = \left(\frac{-3}{5}\right)^{14 + (-3)} = \left(\frac{-3}{5}\right)^{11}
$$

We now have:
$$
\left(\frac{-3}{5}\right)^{11} = \left(\frac{-3}{5}\right)^{3 m+2}
$$

Since the bases on both sides are the same and non-zero, we set the exponents equal:
$$
11 = 3m + 2
$$

Next, solve for $$ m $$ by isolating it:
$$
11 - 2 = 3m \quad \Longrightarrow \quad 9 = 3m
$$
$$
\Rightarrow \quad m = \frac{9}{3} = 3
$$

Thus, the solution is:
$$
m = 3
$$
$$ m = 3 $$

b) \( \left(\frac{-3}{5}\right)^{14} \times\left(\frac{-3}{5}\right)^{-3}=\left(\frac{-3}{5}\right)^{3 m+2} \)

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