Exponents and Division – the Quotient of Powers Property

Exponents and Division – the Quotient of Powers Property exercise

• Explain how to divide $2^5$ by $2^3$.

  Hints

  Remember that $2^5=2\times 2\times 2\times 2\times 2$.

  We can see that $2\times 2\times 2$ in the denominator of $\frac{2\times 2\times 2\times 2 \times 2}{2\times 2 \times 2}$ can cancel out three of the $2$'s in the numerator.

  Solution

  The quotient of powers property says that dividing two powers with the same base is the same as subtracting the exponent of the denominator from the exponent of the numerator and raising the base to that power.

  Generally, this can be stated as $\frac{a^m}{a^n}=a^{(m-n)}$.

  $~$

  Using the quotient of powers property with our example, we have:

  $\frac{2^{\large 5}}{2^{\large 3}}=2^{(5-3)}=2^2$.

  $~$

  Looking at our example even more explicitly, we have that

  ${\large\frac{2^5}{2^3}}=\frac{2\times 2\times 2\times 2 \times 2}{2\times 2 \times 2}$.

  Because we have the factor $2$ five times in the numerator and three times in the denominator we can cancel out three $2$'s in the denominator with three $2$'s in the numerator, giving us

  $\frac{2\times 2\times 2\times 2 \times 2}{2\times 2 \times 2}=2^2$.

  $~$

  We can also come to the same conclusion by dividing $2^5=32$ by $2^3=8$:

  $\frac{32}8=4=2^2$.

• Determine the number of hard drives AA-RNO needs to store an entire copy of the internet.

  Hints

  Keep the quotient of powers property in mind: subtract the exponents and raise the base to this power.

  Here is an example:

  Remember to subtract the exponent of the denominator from the exponent of the numerator... the order matters here!

  Solution

  the quotient of powers property says that we should subtract the exponents and raise the common base to the result: $\frac{a^m}{a^n}=a^{(m-n)}$

  $~$

  So, if we have to divide the total brontobytes we need, $10^{25}$, by any decimal power with base 10, we only have to subtract the exponents and raise 10 to the resulting power.

  $~$

  For our hard drive options, we have:

  $\frac{10^{\large 25}}{10^{\large 15}}=10^{(25-15)}=10^{10}$

  and as well

  ${\large\frac{10^{25}}{10^{18}}}=10^{(25-18)}=10^{7}$

  $~$

  Pay attention to the order of subtraction: You have to subtract the exponent of the denominator from the exponent of the numerator, and not the other way around!

• Decide which division the quotient of powers property can be used.

  Hints

  This power has basis $a$ and exponent $m$.

  Keep in mind that the basis must match... not the exponents.

  Solution

  We can apply the quotient of powers property to powers which have the same base. Then we only have to subtract the exponent of the denominator from the exponent of the numerator and raise the base to the result:

  • ${\large\frac{10^{22}}{10^2}}=10^{(22-2)}=10^{20}$
  • ${\large\frac{10^{22}}{10^{12}}}=10^{(22-12)}=10^{10}$
  • ${\large\frac{10^{22}}{10^3}}=10^{(22-3)}=10^{19}$
  • ${\large\frac{3^{17}}{3^5}}=3^{(17-5)}=3^{12}$
  • ${\large\frac{3^{17}}{3^{10}}}=3^{(17-10)}=3^{7}$
  • ${\large\frac{5^{19}}{5^{10}}}=5^{(19-10)}=5^{9}$
  • ${\large\frac{5^{19}}{5^{3}}}=5^{(19-3)}=5^{16}$
  • ${\large\frac{5^{19}}{5^{12}}}=5^{(19-12)}=5^{7}$

• Examine the growth of the internet data volume between 3008 and 8008.

  Hints

  If you have powers with the same basis you can apply the quotient of powers property:

  Subtract the exponent of the denominator from the exponent of the numerator, not the other way round.

  Let's have a look at an example:

  Solution

  To get the growth $g$ of the internet volume between 8008 and 3008, we have to solve the equation

  $g\times 10^{12}=10^{25}$.

  To get the factor $g$, we need to divide by $10^{12}$; so we have

  $g={\large\frac{10^{25}}{10^{12}}}$.

  Since we have two powers with the same basis, we can apply the quotient of powers property and subtract the exponents in the following way:

  $g=10^{(25-12)}=10^{13}$.

  Thus we can see that the internet data volume has grown by the factor of $10^{13}$.

  Indeed, that is amazing growth!

• Describe the quotient of powers property.

  Hints

  Remember that $a^n=\underbrace{a\times ...\times a}_{\text{n times}}$

  In this example, we can see that the two $3$'s in the denominator cancel out with two of the $3$'s in the numerator.

  In this example, nothing in the denominator can cancel out with anything in the numerator.

  Solution

  This is the formula for the quotient of powers property.

  Let's talk about what it means. This property assists with simplifying powers, but only if they have the same basis.

  When dividing powers with the same basis, you can simply subtract the exponent of the denominator from the exponent of the numerator and raise the common basis to the result.

• Carry out the following divisions.

  Hints

  Keep the common basis and subtract the exponents.

  Pay attention to the order of subtraction. If you change the order of subtraction you'll get a negative exponent. This is also possible but wrong, for instance, for the following example:

  The resulting exponent can also be zero.

  $a^0=1$, as long as $a\neq 0$.

  Solution

  If you have to divide two powers with the same basis, you only have to subtract the exponents in the right order: subtract the exponent of the denominator from the exponent of the numerator. The basis of the result stays the same.

  • ${\large\frac{9^{12}}{9^{11}}}=9^{(12-11)}=9^1=9$
  • ${\large\frac{3^{35}}{3^{33}}}=3^{(35-33)}=3^2=9$
  • ${\large\frac{5^{5}}{5^{2}}}=5^{(5-2)}=5^3=125$
  • ${\large\frac{4^{8}}{4^{8}}=4^{(8-8)}}=4^0=1$
  • ${\large\frac{6^{13}}{6^{10}}}=6^{(13-10)}=6^3=216$
  • ${\large\frac{7^{43}}{7^{21}}}=7^{(43-21)}=7^{22}$