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- https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(Ellis_and_Burzynski)/03%3A_Basic_Operations_with_Real_Numbers/3.07%3A_Negative_ExponentsThen, since \(x^{-3}\) and \(\dfrac{1}{x^3}\) are both reciprocals of \(x^3\) and a real number can have only one reciprocal, it must be that \(x^{-3} = \dfrac{1}{x^3}\). In a fraction, a factor can b...Then, since \(x^{-3}\) and \(\dfrac{1}{x^3}\) are both reciprocals of \(x^3\) and a real number can have only one reciprocal, it must be that \(x^{-3} = \dfrac{1}{x^3}\). In a fraction, a factor can be moved from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent. \dfrac{24 a^{7} b^{9}}{2^{3} a^{4} b^{-6}}=\dfrac{24 a^{7} b^{9}}{8 a^{4} b^{-6}} &=3 a^{7-4} b^{9-(-6)} \\
- https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/07%3A_Exponents_and_Scientific_Notation/7.02%3A_New_Page/7.2.6%3A_Practice_with_Rational_BasesWe also extended these rules to make sense of negative exponents as repeated factors of the reciprocal of the base, as well as defining a number to the power of 0 to have a value of 1. \[x^{n}\cdot x^...We also extended these rules to make sense of negative exponents as repeated factors of the reciprocal of the base, as well as defining a number to the power of 0 to have a value of 1. \[x^{n}\cdot x^{m}=x^{n+m},\qquad (x^{n})^{m}=x^{n\cdot m},\qquad\frac{x^{n}}{x^{m}}=x^{n-m},\qquad x^{-n}=\frac{1}{x^{n}},\quad\text{and}\quad x^{0}=1\nonumber\] For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/02%3A_Fractions_and_Mixed_Numbers/2.02%3A_Multiplying_and_Dividing_Fractions_and_Mixed_Numbers/2.2.02%3A_Dividing_Fractions_and_Mixed_NumbersThe fractional parts of this answer and the original mixed number are reciprocals, but in order to find the reciprocal of the entire number, you must write the mixed number as an improper fraction bef...The fractional parts of this answer and the original mixed number are reciprocals, but in order to find the reciprocal of the entire number, you must write the mixed number as an improper fraction before interchanging numerator and denominator. An easy way to remember how to divide fractions is the phrase “keep, change, flip”. This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.
- https://math.libretexts.org/Courses/Fullerton_College/Math_100%3A_Liberal_Arts_Math_(Claassen_and_Ikeda)/10%3A_Appendix/10.02%3A_Fractions_and_Mixed_Numbers/10.2.02%3A_Multiplying_and_Dividing_Fractions_and_Mixed_Numbers/10.2.2.02%3A_Dividing_Fractions_and_Mixed_NumbersThe fractional parts of this answer and the original mixed number are reciprocals, but in order to find the reciprocal of the entire number, you must write the mixed number as an improper fraction bef...The fractional parts of this answer and the original mixed number are reciprocals, but in order to find the reciprocal of the entire number, you must write the mixed number as an improper fraction before interchanging numerator and denominator. An easy way to remember how to divide fractions is the phrase “keep, change, flip”. This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.
- https://math.libretexts.org/Courses/Santiago_Canyon_College/HiSet_Mathematica_(Lopez)/17%3A_Operaciones_Basicas_con_Numeros_Reales/17.07%3A_Exponentes_negativosEso también lo sabemos\(x^3 \cdot \dfrac{1}{x^3} = 1\). (Ver problema 6 anterior.) Así,\(x^3\) y también\(\dfrac{1}{x^3}\) son recíprocos. Entonces, dado que\(x^{-3}\) y\(\dfrac{1}{x^3}\) son ambos re...Eso también lo sabemos\(x^3 \cdot \dfrac{1}{x^3} = 1\). (Ver problema 6 anterior.) Así,\(x^3\) y también\(\dfrac{1}{x^3}\) son recíprocos. Entonces, dado que\(x^{-3}\) y\(\dfrac{1}{x^3}\) son ambos recíprocos de\(x^3\) y un número real puede tener sólo un recíproco, debe ser eso\(x^{-3} = \dfrac{1}{x^3}\). \ dfrac {24 a^ {7} b^ {9}} {2^ {3} a^ {4} b^ {-6}} =\ dfrac {24 a^ {7} b^ {9}} {8 a^ {4} b^ {-6}} &=3 a^ {7-4} b^ {9- (-6)}\
- https://math.libretexts.org/Bookshelves/PreAlgebra/Prealgebra_(Arnold)/04%3A_Fractions/4.04%3A_Dividing_Fractions\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \t...\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ = - \frac{(2 \cdot 3) \cdot (5 \cdot 11)}{(5 \cdot 7) \cdot (3 \cdot 11)} ~ & \textcolor{red}{ \text{ Factor numerators and denominators.}} \\ = - \frac{2 \cdot \cancel{3} \cdot \cancel{5} \cdot \cancel{11}}{ \cancel{5} \cdot 7 …
- https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_Math_with_Business_Apps/02%3A_Fractions/2.03%3A_Dividing_Fractions\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \t...\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ = - \frac{(2 \cdot 3) \cdot (5 \cdot 11)}{(5 \cdot 7) \cdot (3 \cdot 11)} ~ & \textcolor{red}{ \text{ Factor numerators and denominators.}} \\ = - \frac{2 \cdot \cancel{3} \cdot \cancel{5} \cdot \cancel{11}}{ \cancel{5} \cdot 7 …
- https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_PreAlgebra/02%3A_Fractions/2.03%3A_Dividing_Fractions\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \t...\[ \begin{aligned} - \frac{6}{35} \div \frac{33}{55} = - \frac{6}{35} \cdot \frac{55}{33} ~ & \textcolor{red}{ \text{ Invert the divisor (second number).}} \\ = - \frac{6 \cdot 55}{35 \cdot 33} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ = - \frac{(2 \cdot 3) \cdot (5 \cdot 11)}{(5 \cdot 7) \cdot (3 \cdot 11)} ~ & \textcolor{red}{ \text{ Factor numerators and denominators.}} \\ = - \frac{2 \cdot \cancel{3} \cdot \cancel{5} \cdot \cancel{11}}{ \cancel{5} \cdot 7 …
