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If (frac(5)(3))^(-5) xx (frac(5)(3))^(11...

If `(frac(5)(3))^(-5) xx (frac(5)(3))^(11)=(frac(5)(3))^(8x),` then x = ?

A

(a) `frac(-1)(2)`

B

(b) `frac(-3)(4)`

C

(c) `frac(3)(4)`

D

(d) `frac(4)(3)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the equation \((\frac{5}{3})^{-5} \times (\frac{5}{3})^{11} = (\frac{5}{3})^{8x}\), we can follow these steps: ### Step 1: Apply the exponent multiplication rule According to the exponent rule, when we multiply two powers with the same base, we add the exponents. Therefore, we can rewrite the left side of the equation: \[ (\frac{5}{3})^{-5} \times (\frac{5}{3})^{11} = (\frac{5}{3})^{-5 + 11} \] ### Step 2: Simplify the exponent Now, simplify the exponent on the left side: \[ -5 + 11 = 6 \] So, we have: \[ (\frac{5}{3})^{6} = (\frac{5}{3})^{8x} \] ### Step 3: Set the exponents equal to each other Since the bases are the same, we can set the exponents equal to each other: \[ 6 = 8x \] ### Step 4: Solve for \(x\) Now, we need to isolate \(x\). To do this, divide both sides of the equation by 8: \[ x = \frac{6}{8} \] ### Step 5: Simplify the fraction Now simplify \(\frac{6}{8}\): \[ x = \frac{3}{4} \] ### Final Answer Thus, the value of \(x\) is: \[ x = \frac{3}{4} \] ---
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