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$Express the following fraction in simplest form, only using positive exponents. (6x^(2)t)/((-2x^(-3)t^(4))^(5)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{6 x^{2} t}{\left(-2 x^{-3} t^{4}\right)^{5}}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

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

\newline

\frac{6 x^{2} t}{\left(-2 x^{-3} t^{4}\right)^{5}}

\newline

Answer:

Distribute Exponent of $5$ : We need to simplify the fraction $\frac{6x^{2}t}{(-2x^{-3}t^{4})^{5}}$ . First, let's simplify the denominator by distributing the exponent of $5$ to each factor inside the parentheses.
Apply Power of Power Rule: Apply the power of a power rule: a*b)^n = a^n * b^n\. In this case, we have \$\left(-2x^{(-3)}t^{(4)}\right)^5 = (-2)^5 * \$x^{(-3)}^ $5$ * $t^{(4)}$ ^ $5$ \.
Calculate Parts Separately: Calculate each part separately: $(-2)^5 = -32$ , $(x^{-3})^5 = x^{-15}$ , and $(t^{4})^5 = t^{20}$ .
Rewrite Denominator: Now, rewrite the denominator with the calculated values: $(-2x^{-3}t^{4})^{5} = -32x^{-15}t^{20}$ .
Divide Numerator by Denominator: The fraction now looks like this: $(6x^{2}t)/(-32x^{-15}t^{20})$ . Next, we will divide the numerator by the denominator by subtracting the exponents of like bases and dividing the coefficients.
Divide Coefficients: Divide the coefficients: $\frac{6}{-32} = -\frac{3}{16}$ . Subtract the exponents of $x$ : $x^{(2 - (-15))} = x^{17}$ . Subtract the exponents of $t$ : $t^{(1 - 20)} = t^{-19}$ .
Subtract Exponents: Combine the results: $-\frac{3}{16} \times x^{17} \times t^{-19}$ . Since we want only positive exponents, we need to move $t^{-19}$ to the denominator.
Combine Results: The final simplified expression with positive exponents is: $(-3x^{17})/(16t^{19})$ .

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