Solutions

1. A (Written Solution)

   The negative exponent rule says that a factor with a negative exponent can be moved to the other part of the fraction (top to bottom or bottom to top) and then have a positive exponent.

   In this example, \({\text{x}}^{-2}\) has a negative exponent of \(-2\), but it doesn’t look like a fraction. In cases like this, you must remember that anything divided by 1 is still itself. For example \(\frac{3}{1}=3\) or \(12=\frac{12}{1}\)

   We can apply this same principle here.

   \({\text{x}}^{-2}=\frac{{\text{x}}^{-2}}{1}\)

   Now it looks like a fraction.

   The negative exponent rule now says we can do the following:

   \({\text{x}}^{-2}=\frac{{\text{x}}^{-2}}{1} \rightarrow \frac{1}{{\text{x}}^{2}}\)

   Our final answer is \(\frac{1}{{\text{x}}^{2}}\)



2. C


3. \({\text{y}}^{6}\)(Written Solution)

   Version 1: Use the negative exponent rule and then quotient rule

   First, we apply the negative quotient rule that says as long as all the factors are being multiplied or divided together (no addition or subtraction) then we can move a factor with a negative exponent to the opposite side of a fraction and change the exponent to a positive.

   Applying this we get:

   \(\frac{{\text{y}}^{-7}}{{\text{y}}^{-13}}=\frac{{\text{y}}^{13}}{{\text{y}}^{7}}\)

   Next, we apply the quotient rule that says as long as all the factors are being multiplied or divided (no addition or subtraction), and as long as factors have the same base, we can subtract the exponent of the factor in the denominator (bottom of the fraction) from the exponent of the factor in the numerator (top of the fraction).

   \(\frac{{\text{y}}^{13}}{{\text{y}}^{7}}={\text{y}}^{\left ( 13-7 \right )}={\text{y}}^{6}\)

   Version 2: Use the quotient rule and then the negative exponent rule

   First, we apply the quotient rule. The quotient rule says that as long as the bases are the same, we can subtract the exponent of the factor in the denominator from the exponent of the factor in the numerator.

   Applying this we get:

   \(\frac{{\text{y}}^{-7}}{{\text{y}}^{-13}}={\text{y}}^{\left ( -7-\left ( -13 \right ) \right )}\)

   \(\frac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})}\)

   Subtracting a negative is the same as addition, so the exponent of \({\text{y}}\) is:

   \((-7-(-13)) = -7 + 13\)

   \(\frac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})} = {\text{y}}^{(-{\color{Blue} 7}+{\color{DarkGreen} 13})}\)

   So \(-7 + 13 = 6\) and our final answer is \({\text{y}}^{6}\).

   \(\frac{{\text{y}}^{{\color{Blue} -7}}}{{\text{y}}^{{\color{DarkGreen} -13}}} = {\text{y}}^{({\color{Blue} -7}-\color{DarkGreen}(-13)\color{black})} = {\text{y}}^{(-{\color{Blue} 7}+{\color{DarkGreen} 13})} = {\color{Red} {\text{y}}^{6}}\)



4. \(\frac{{\text{x}}^{5}}{{\text{m}}^{3}}\)


5. \(\frac{{\text{a}}^{8}}{{\text{b}}^{5}}\) (Solution Video | Transcript)


6. \(\frac{\text{y}}{{\text{x}}^{8}{\text{z}}^{2}}\) (Solution Video | Transcript)