Express the following fraction in simplest form, only using positive exponents. ((3a^(-1)t^(3))^(-2))/(12 a) Answer:

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Express the following fraction in simplest form, only using positive exponents.  ((3a^(-1)t^(3))^(-2))/(12 a) Answer:

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Apply Negative Exponent Rule: Apply the negative exponent rule to the numerator. The negative exponent rule states that x^(-n) = 1/(x^n). We will apply this rule to the entire numerator. ((3a^(-1)t^(3))^(-2)) = (1/(3a^(-1)t^(3))^2)Simplify Expression Inside Parentheses: Simplify the expression inside the parentheses before applying the exponent. We need to convert the negative exponent inside the parentheses to a positive exponent by moving it to the denominator. (1/(3a^(-1)t^(3))^2) = (1/(3*(1/a)*t^(3))^2)Continue Simplifying Inside Parentheses: Continue simplifying the expression inside the parentheses. Now we can simplify the expression inside the parentheses by multiplying the terms. (1/(3*(1/a)*t^(3))^2) = (1/(3/a*t^(3))^2)Apply Exponent to Terms: Apply the exponent to the terms inside the parentheses. When we raise a product to an exponent, we raise each factor to that exponent. (1/(3/a*t^(3))^2) = (1/((3^2/a^2)*t^(6)))Simplify Expression After Exponent: Simplify the expression after applying the exponent. Now we can simplify the expression by calculating the exponent of 3 and moving the a term to the numerator. (1/((3^2/a^2)*t^(6))) = (1/(9/a^2*t^(6)))Multiply by Reciprocal: Multiply the fraction by the reciprocal of the denominator. To get rid of the fraction in the denominator, we multiply by the reciprocal. (1/(9/a^2*t^(6))) = (a^2/(9*t^(6)))Divide by Denominator: Divide the entire expression by the denominator outside the parentheses, which is 12a. Now we divide the expression we have by 12a. (a^2/(9*t^(6)))/(12a) = (a^2/(9*t^(6)*12a))Simplify by Canceling Factors: Simplify the expression by canceling out common factors. We can cancel out an 'a' from the numerator and denominator and simplify the constants. (a^2/(9*t^(6)*12a)) = (a/(9*t^(6)*12))Simplify Constants in Denominator: Simplify the constants in the denominator. Now we simplify the constants 9 and 12 by dividing them. (a/(9*t^(6)*12)) = (a/(108*t^(6)))

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{\left(3 a^{-1} t^{3}\right)^{-2}}{12 a}$ $\newline$ Answer:

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Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{\left(3 a^{-1} t^{3}\right)^{-2}}{12 a}$ $\newline$ Answer: