Questions: Express the following fraction in simplest form, only using positive exponents. 5(t^(-4) x^(-5))^(-1) / (3 t^4 x^(-1))

Express the following fraction in simplest form, only using positive exponents.

5(t^(-4) x^(-5))^(-1) / (3 t^4 x^(-1))
Transcript text: Express the following fraction in simplest form, only using positive exponents. \[ \frac{5\left(t^{-4} x^{-5}\right)^{-1}}{3 t^{4} x^{-1}} \]
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Solution

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Solution Steps

Step 1: Simplify the numerator

The numerator is 5(t4x5)15\left(t^{-4} x^{-5}\right)^{-1}. Using the power of a power rule (am)n=amn(a^m)^n = a^{m \cdot n}, we simplify the expression inside the parentheses: (t4x5)1=t4x5. \left(t^{-4} x^{-5}\right)^{-1} = t^{4} x^{5}. Thus, the numerator becomes: 5t4x5. 5 \cdot t^{4} x^{5}.

Step 2: Rewrite the denominator

The denominator is 3t4x13 t^{4} x^{-1}. Rewrite x1x^{-1} as 1x\frac{1}{x}: 3t4x1=3t41x. 3 t^{4} x^{-1} = 3 t^{4} \cdot \frac{1}{x}.

Step 3: Combine the numerator and denominator

Now, divide the numerator by the denominator: 5t4x53t41x. \frac{5 t^{4} x^{5}}{3 t^{4} \cdot \frac{1}{x}}. Simplify by canceling t4t^{4} in the numerator and denominator: 5x531x. \frac{5 x^{5}}{3 \cdot \frac{1}{x}}. Multiply by the reciprocal of 1x\frac{1}{x}: 5x5x3=5x63. \frac{5 x^{5} \cdot x}{3} = \frac{5 x^{6}}{3}.

Final Answer

5x63\boxed{\frac{5 x^{6}}{3}}

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