Exponents and scientific notation
Integer exponents
Properties of exponents
The product rule: multiplication of like bases
$a^m\cdot a^n=a^{m+n}$
The quotient rule: division of like bases
$\dfrac{a^m}{a^n}=a^{m-n}$
The power rule:
$(a^m)^n=a^{mn}$
Power of a product:
$(ab)^m=a^m\cdot b^m$
Power of a quotient:
$\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$
The zero exponent:
$a^0=1$
Proof:
We can write
$a^0=a^{n-n}$
Applying the quotient rule to the right-hand side,
$a^0=\dfrac{a^n}{a^{n}}=1$
The negative exponent:
$a^{-n}=\dfrac{1}{a^n}$
$\left(\dfrac{a}{b}\right )^{-n}=\left(\dfrac{b}{a}\right )^{n}$
$\left(\dfrac{a}{b}\right )^{-n}=\dfrac{a^{-n}}{b^{-n}}$
Using these properties, we can simplify expressions involving exponents.
Example 1: Simplify, $\left (8a^5b^4 \right )^3$.
Solution:
Applying the power of a product rule and the power rule,
$\begin{align*} \left (8a^5b^4 \right )^3 &= 8^3(a^5)^3(b^4)^3\\ &= 8^3a^{5\cdot 3}b^{4\cdot 3}\\ &=512a^{15}b^{12}\\ \end{align*}$
Example 2: Simplify, $\left (2b^3c^4 \right )^{-4}$.
Solution:
Applying the negative-exponent rule,
$\begin{align*} \left (2b^3c^4 \right )^{-4} &= \dfrac{1}{ \left (2b^3c^4 \right )^{4}} \\ &=\dfrac{1}{16\,b^{12}\,c^{16}}\\ \end{align*}$
Example 3: Simplify, $(2x^3yb^2)^3(5x^2y^5)$
Solution:
$\begin{align*} \left(2x^3yb^2\right)^3 \left (5x^2y^5\right) &= \left(8x^9y^3b^6\right)\left (5x^2y^5\right)\\ &=40x^{11}y^8b^6 \\ \end{align*}$
Example 4: Simplify, $\left(\dfrac{4u^8b^7z}{3u^3b^6z^3}\right)^2$
Solution:
First simplify what is within the parenthesis. Then apply the power rule.
$\begin{align*} \left(\dfrac{4u^8b^7z}{3u^3b^6z^3}\right)^2 &= \left(\dfrac{4u^5b}{3z^2}\right)^2\\ &=\dfrac{16u^{10}b^2}{9z^4} \end{align*}$
Example 5: Simplify $\left(\dfrac{4u^8b^3\,z^3}{3u^3b^6z}\right)^{-3}$
Solution:
Take the reciprocal of the expression within the parenthesis, then apply the power rule.
$\begin{align*} \left(\dfrac{4u^8b^3z^3}{3u^3b^6z}\right)^{-3} &= \left(\dfrac{3u^3b^6z}{4u^8b^3\,z^3}\right)^{3}\\ &= \left(\dfrac{3\,b^3}{4u^5\,z^2}\right)^{3}\\ &= \dfrac{27\,b^9}{64\,u^{15}\,z^6} \end{align*}$
Example 6: Simplify $8\,z^{-7}\,y^2\cdot 5\,z^3\,a^2\,y^{-5}$
Solution:
$\begin{align*} 8\,z^{-7}\,y^2\cdot 5\,z^3\,a^2\,y^{-5} &= 8\cdot 5 \,z^{-7+3}\,y^{2-5}a^2\\ &=40\,z^{-4}y^{-3}a^2\\ &=\dfrac{40\,a^2}{z^4\,y^3} \end{align*}$
Example 7: Simplify $\dfrac{7x^{-8}\,y^{11}}{b^5\,x^5\,y^4}$
Solution:
$\begin{align*} \dfrac{7x^{-8}\,y^{11}}{b^5\,x^5\,y^4} &= \dfrac{7\,y^{11-4}}{b^5\,x^{5+8}}\\ &=\dfrac{7\,y^7}{b^5\,x^{13}}\\ \end{align*}$
Scientific notation
Scientific notation is a convenient way to write a very small or a very large number. It is written in the form,
$a\times 10^n$, where $n$ is a positive or negative integer and $1\le a \lt 10$.
Example 1: Write in scientific notation, $7600000000$
Answer: The number is greater than $10$ and has no decimal point. So, put a decimal point after the first digit, $7$. Now, count how many digits are after the decimal point — there are $9$. This number is $n$, so $n=9$.
Therefore, $7600000000=7.600000000\times 10^9$
Ignore all the zeros after the last non-zero digit, $6$, since they have no value. You get,
$7600000000=7.6\times 10^9$
Example 2: Write in scientific notation, $0.000000065$
Answer: The number is less than $1$. So, move the decimal point to the right and put it after the first non-zero digit, $6$. Since you moved the decimal point to the right, $n$ will be negative. Now, count the number of places the decimal point has moved, which is $8$. The negative of this number is $n$, so $n=-8$.
Therefore, $0.000000065=6.5\times 10^{-8}$
Note that I ignored the zeros before the digit $6$, since they have no value.
Example 3: Write the following number in standard notation, $4.82\times 10^{-6}$
Answer: Since the exponent is negative, move the decimal point to the left. The exponent is $-6$, so move the point $6$ places to the left. Since there is only one digit to the left of the decimal point, add $5$ zeros for the remaining places.
So, $4.82\times 10^{-6}=0.00000482$
Note that I have put a zero to the left of the decimal point, which is optional.
Example 4: Write $6.25\times 10^5$ in standard notation.
Answer: Since the exponent is positive, move the decimal point five places to the right, adding zeros if there are not enough places. Remove the decimal point if there are no non-zero digits after it.
So, $6.25\times 10^5=625000$