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$Express the following fraction in simplest form, only using positive exponents. (12p^(-10))/((-4p^(-2))^(4)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{12 p^{-10}}{\left(-4 p^{-2}\right)^{4}}$ $\newline$ Answer:

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

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

\newline

\frac{12 p^{-10}}{\left(-4 p^{-2}\right)^{4}}

\newline

Answer:

Simplify denominator: Simplify the denominator. $\newline$ We have the denominator as $(-4p^{-2})^4$ . When raising a power to a power, we multiply the exponents. Also, the negative sign raised to an even power will become positive. $\newline$ $((-4p^{-2})^4) = (-4)^4 \times (p^{-2})^4$ $\newline$ $= 256 \times p^{-8}$
Rewrite with positive exponents: Rewrite the expression with positive exponents. $\newline$ We can rewrite $p^{-8}$ as $1/p^8$ to make the exponent positive. $\newline$ $256 \times p^{-8} = 256/p^8$
Divide numerator by denominator: Divide the numerator by the denominator. $\newline$ Now we divide the original numerator by the simplified denominator. $\newline$ $(12p^{-10}) / (256/p^8)$ $\newline$ = $(12p^{-10}) \times (p^8/256)$
Simplify expression: Simplify the expression. $\newline$ We can multiply the $p$ terms by adding the exponents and divide the coefficients. $\newline$ $(12 \times p^{-10}) \times (p^8/256)$ $\newline$ = $(12/256) \times p^{-10+8}$ $\newline$ = $(12/256) \times p^{-2}$
Reduce coefficient fraction: Reduce the coefficient fraction. $\newline$ The fraction $\frac{12}{256}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ . $\newline$ $\left(\frac{12}{256}\right) = \frac{3}{64}$
Rewrite with positive exponents: Rewrite the final expression with positive exponents. $\newline$ We can rewrite $p^{-2}$ as $1/p^2$ to make the exponent positive. $\newline$ $(3/64) \cdot p^{-2} = (3/64) \cdot (1/p^2)$
Combine coefficients and variables: Combine the coefficients and the variables. $\newline$ Now we combine the coefficients and the variables to express the final answer. $\newline$ $(\frac{3}{64}) \times (\frac{1}{p^2}) = \frac{3}{64p^2}$