Zero and Negative Exponents

Zero and Negative Exponents exercise

• Decide what $10^{-5}$ stands for.

  Hints

  For example, $10^3=10\times 10\times 10$.

  You multiply two decimal powers by adding their exponents:

  $10^7\times 10^{-5}=10^{7-5}=10^2$.

  Here you see how to handle a negative exponent for $x\neq 0$.

  Solution

  Mr. Magic shrinks the picture by a decimal power: $10^{-5}$. How can we see what this negative decimal power is trying to express? Using what we know about multiplying and dividing powers with exponents, we can see that:

  $10^{\large -5}=10^{\large 0-5}=10^{\large 0}\div 10^{\large 5}=1\div 10^{\large 5}=\frac{1}{10^{\large 5}}=\frac{1}{100000}$.

• Explain why $2^{-4} = \frac1{2^4}$ is true.

  Hints

  Here is the rule for dividing powers with the same basis.

  Each power with a zero exponent is equal to $1$ for $x\neq 0$:

  $\large x^0=1$

  Solution

  We start with the fraction $\frac1{2^{\large 4}}$.

  Since each power with a zero exponent is equal to $1$, we have that $2^{0}=1$. So $1$ over $2$ to the power of $4$ is equal to $2^{0}\div 2^{4}$.

  Since $2^{4}$ and $2^{0}$ have the same basis, we can subtract their exponents by the rule for dividing powers to get $2^{0-4}$.

  We then have that $\frac1{2^{\large 4}}=2^{-4}$.

• Examine the following powers with negative exponents.

  Hints

  Keep the general formula for negative exponents in mind.

  Count the number of zeros in the denominator.

  The number of zeros in the denominator is the same as the absolute value of the exponent.

  Solution

  In general we have $x^{-a}=\frac1{x^{\large a}}$, where $x\neq 0$.

  So if we have $x=10$, we get:

  • $10^{-3}$$=\frac1{10^{\large 3}}=\frac1{1000}$

  • $10^{-1}$$=\frac1{10^{\large 1}}=\frac1{10}$

  • $10^{-6}$$=\frac1{10^{\large 6}}=\frac1{1000000}$

  • $10^{-7}$$=\frac1{10^{\large 7}}=\frac1{10000000}$

  Notice that the number of zeros in the denominator is the same as the absolute value of the exponent.

• Decide the power of the enlarging potion.

  Hints

  Find the shrinking factor of the spell.

  Divide the size of the shrunken castle by the size of the original castle.

  The enlarging factor is the reciprocal of the shrinking factor.

  If you write the shrinking factor as $1$ over a decimal power with a positive exponent, you can find the reciprocal directly.

  Solution

  First let's find the shrinking factor of the spell. To do this, we divide the resulting size of the castle $2$ by the original size $200$:

  $\frac2{200}=\frac1{100}=\frac1{10^{\large 2}}$.

  Writing this as a decimal power with a negative exponent we get $10^{-2}$.

  So to undo the shrinking, we have to multiply by $10^2=100$.

  Let's check: $2\times 100=200$ $\surd$.

  So $10^2=100$ is the enlarging factor we need for the potion.

• Explain how to write $x^{-a}$ as a fraction.

  Hints

  An example of a negative exponent is $2^{-4}=\frac1{2^{\large 4}}$.

  An example with $x$ as the basis is $x^{-4}=\frac1{x^{\large 4}}$.

  Remember that division by zero isn't allowed.

  Solution

  We've already seen that

  $10^{-5}=\frac1{10^{\large 5}}$

  We can prove that $2^{-4}=\frac1{2^{\large 4}}$ using $2^0=1$:

  $\begin{array}{lcr} \frac1{2^{\large 4}}&=&\frac{2^{\large 0}}{2^{\large 4}}\\ &=&2^{0-4}\\ &=&2^{-4} \end{array}$

  Similarly we have $x^{-4}=\frac1{x^{\large 4}}$.

  In general, we have

  $x^{-a}=\frac1{x^{\large a}}$.

  As long as the basis does not equal zero, if we have a power with a negative exponent we can also write it as a fraction with $1$ in the numerator and the power with the same basis to the absolute value of the exponent in the denominator.

• Identify the powers resulting from the calculations shown.

  Hints

  In general $\large 10^{-a}=\frac1{10... 0}$, where the number of zeros in the denominator is the same as the absolute value of the exponent.

  If you divide $1$ by a decimal power you get the resulting decimal number, $0.0...01$, where the position of $1$ after the comma is the same as the exponent.

  Solution

  Let's start with the fraction $\frac{10^{\large 3}}{10^{\large 6}}$.

  Since we have two powers with the same basis, we can subtract the exponents to get $\frac{10^{\large 3}}{10^{\large 6}}=10^{\large 3-6}=10^{\large -3}$.

  Next we can write the power as a fraction $10^{\large -3}=\frac1{1000}$. To get the corresponding decimal number we write the $1$ in the third position after the comma:

  $\frac{10^{\large 3}}{10^{\large 6}}=\frac1{1000}=0.001$.

  $~$

  Next we multiply two powers with the same exponents: $\left(\frac12\right)^{\large -4}\times 20^{\large -4}$. First we multiply both base numbers together to get $\frac12\times 20=10$.

  The resulting power has the same exponent as both factors. So we have $\left(\frac12\right)^{\large -4}\times 20^{\large -4}=10^{\large -4}$.

  Again we write this power as a fraction and after as a decimal number: $10^{\large -4}=\frac1{10^{\large 4}}=\frac1{10000}=0.0001$

  $~$

  We handle the last, $8^{\large -3}\times125^{\large -3}$, in a similar manner.

  We have $8\times 125=1000=10^{\large 3}$, so we can conclude that $8^{\large -3}\times125^{\large -3}=\left(10^3\right)^{-3}=10^{\large -9}$.

  Writing this result as a fraction and a decimal number, we get $10^{\large -9}=\frac1{10^{\large 9}}=\frac1{1000000000}=0.000000001$.