Fully simplify using only positive exponents. (45x^(2)y^(5))/(27x^(6)y^(3)) Answer:

Factorization: Factor both the numerator and the denominator to reveal common factors. 45 can be factored into 3 * 3 * 5, and 27 can be factored into 3 * 3 * 3. So, (45x^(2)y^(5))/(27x^(6)y^(3)) becomes ((3 * 3 * 5)x^(2)y^(5))/((3 * 3 * 3)x^(6)y^(3)).Common Factor Cancellation: Cancel out the common factors of 3 * 3 from the numerator and the denominator. This leaves us with (5x^(2)y^(5))/(3x^(6)y^(3)).Quotient Rule for Exponents: Apply the quotient rule for exponents, which states that a^(m)/a^(n) = a^(m-n) when m > n. For x^(2)/x^(6), we subtract the exponents: 2 - 6 = -4, which gives us x^(-4). For y^(5)/y^(3), we subtract the exponents: 5 - 3 = 2, which gives us y^(2). So, we have (5x^(-4)y^(2))/(3).Adjusting Negative Exponent: Since we want only positive exponents, we move x^(-4) to the denominator to make the exponent positive. This gives us (5y^(2))/(3x^(4)).Final Answer: There are no further simplifications possible, so this is the final answer.

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Q. Fully simplify using only positive exponents.

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\frac{45 x^{2} y^{5}}{27 x^{6} y^{3}}

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Answer:

Factorization: Factor both the numerator and the denominator to reveal common factors. $\newline$ $45$ can be factored into $3 \times 3 \times 5$ , and $27$ can be factored into $3 \times 3 \times 3$ . $\newline$ So, $\frac{45x^{2}y^{5}}{27x^{6}y^{3}}$ becomes $\frac{(3 \times 3 \times 5)x^{2}y^{5}}{(3 \times 3 \times 3)x^{6}y^{3}}$ .
Common Factor Cancellation: Cancel out the common factors of $3 \times 3$ from the numerator and the denominator. $\newline$ This leaves us with $(5x^{2}y^{5})/(3x^{6}y^{3})$ .
Quotient Rule for Exponents: Apply the quotient rule for exponents, which states that $a^{m}/a^{n} = a^{m-n}$ when m > n. For $x^{2}/x^{6}$ , we subtract the exponents: $2 - 6 = -4$ , which gives us $x^{-4}$ . For $y^{5}/y^{3}$ , we subtract the exponents: $5 - 3 = 2$ , which gives us $y^{2}$ . So, we have $(5x^{-4}y^{2})/(3)$ .
Adjusting Negative Exponent: Since we want only positive exponents, we move $x^{-4}$ to the denominator to make the exponent positive. $\newline$ This gives us $(\frac{5y^{2}}{3x^{4}})$ .
Final Answer: There are no further simplifications possible, so this is the final answer.