Basic exponents

Exponential notation

Exponential notation simplifies repeated multiplication. Below is an example of an exponent:

$$2^{3}=2\cdot2\cdot2=8$$

In the above expression, \( 2 \) is the base and \( 3 \) is the exponent. You would read \( 2^{3} \) as: “two raised to the power of three,” or “two to the third.” As shown above, \( 2^{3}\), equals two times two times two. Using generic terms \( x^{n} \) is equal to \( 1\) multiplied by \(x\) \( n \) times.

$$x^{n}=1\cdot \overbrace{x\cdot x\cdot x\ldots\cdot x}^{n \; times}$$

This list of powers of two should help you see the pattern.

$$\begin{align*} & 2^{0}=1 & = & 1 \\ & 2^{1}=1\cdot2 & = & 2 \\ & 2^{2}=1\cdot2\cdot2 & = & 4 \\ & 2^{3}=1\cdot2\cdot2\cdot2 & = & 8 \\ & 2^{4}=1\cdot2\cdot2\cdot2\cdot2 & = & 16 \\ & 2^{5}=1\cdot2\cdot2\cdot2\cdot2\cdot2 & = & 32 \end{align*}$$

Can you figure out what \( 2^{6} \) is?

Because the placeholding \( 1 \) does not affect the outcome, it is usually not written; we’re using it here only to help demonstrate the pattern.

You may be stumped as to why \( 2^{0}=1 \). In fact, any number raised to the power of \( 0 \) equals \( 1 \). If you refer to our definition of the notation, \( x^{0} \) is equal to \( 1 \) times \( x \) zero times, so there is no multiplication and we are only left with \( 1 \).

If \( x \) is negative, the notation works the same but we must use parentheses correctly. Note the different in the two expressions below:

$$\begin{align*} (- & 3)^{2} & = & 1 \cdot-3\cdot-3 &=& &&9 \\ - & 3^{2} & = & -1\cdot3^{2}=-1\cdot3\cdot3 &=& &-&9 \end{align*}$$

It is important to to notice how the parentheses affect the outcome. In the first example \( -3 \) is squared. In the second example only 3 is squared, and the negative sign is applied to that result.

Now what if n is negative? For example \( 2^{-3} \). In this case you would simply divide one by two three times.

$$2^{-3}=1/2/2/2=\frac{1}{2\cdot2\cdot2}=\frac{1}{8}$$

Using the generic number \( x \) and \( n \) we have the definition:

$$x^{-n}=\underbrace{\frac{1}{x\cdot x\cdot x\ldots\cdot x}}_{n \; times}$$

Since division is the opposite of multiplication, if your exponent is negative, you divide 1 by x n times. This list of some of the positive and negative powers of three will help show the pattern.

$$\begin{align*} & 3^{-3} & = & \frac{1}{3\cdot3\cdot3} & = & \frac{1}{27} \\ & 3^{-2} & = & \frac{1}{3\cdot3} & = & \frac{1}{9} \\ & 3^{-1} & = & \frac{1}{3} & = & \frac{1}{3} \\ & 3^{0} & = & 1 & = & 1 \\ & 3^{1} & = & 1\cdot3 & = & 3 \\ & 3^{2} & = & 1\cdot3\cdot3 & = & 9 \\ & 3^{3} & = & 1\cdot3\cdot3\cdot3 & = & 27 \end{align*}$$

Square roots

Any number raised to the power of 2 is said to be ‘squared.’ Taking a square root of a number reverses the process or ‘squaring.’ For Example:

$$4^{2}=16 \quad \sqrt{16}=4$$

Here’s an example of solving a basic equation for x:

$$\begin{align*} & x^{2}=81 \\ & \rightarrow \sqrt{x^{2}}=\sqrt{81} \\ & \rightarrow x=9 \\ & (\text{since } 9^{2}=81) \end{align*}$$

Scientific notation

It is not practical to write out really long numbers, like \( 42,000,000,000,000,000,000,000,000 \).

Instead we use scientific notation to compress all the zeros. In scientific notation, this number becomes \( 4.2\cdot10^{25} \).

Both numbers are mathematically the same, but expressing the number in exponential notation conserves space by eliminating the need to write out all the zeros.

If you count the digits to the right of \( 4 \) you’ll come up with \( 25 \), and this is no coincidence. \( 10^{25}=10,000,000,000,000,000,000,000,000 \) so when you multiply \( 4.2 \) by \( 10^{25} \) you get \( 42,000,000,000,000,000,000,000,000 \).

Take a look at these examples and try to spot the general pattern.

$$\begin{align*} 175,020,000,000 & = & 1.7502 & \cdot 10^{11} \\ 23,000 & = & 2.3 & \cdot 10^{4} \\ 1,000,000,000 & = & 1 & \cdot 10^{9} \\ 5,056,700,000,000,000 & = & 5.0567 & \cdot 10^{15} \end{align*}$$

You can also use scientific notation on very small numbers. The method is the similar but you’re working backwards and so negative powers of ten are used. Here are some examples:

$$\begin{align*} 0.00001 & = & 1 & \cdot 10^{-5} \\ 0.0000007708 & = & 7.708 & \cdot 10^{-7} \\ 0.000001005 & = & 1.005 & \cdot 10^{-6} \end{align*}$$

The principle of scientific notation is that multiplication or division by 10 moves the decimal place in a number. Refer to the study guide on decimals for a more detailed discussion on this.

You have likely seen numbers in scientific notation before on your calculator. Since calculator screens are not large enough to display very large numbers, calculators use scientific notation to display oversize numbers. For example, the BA II Plus can only fit 10 digits on the screen, anything larger must be displayed using scientific notation.

If you calculate \( 1,000,000^{2} \) on this model the screen displays: \( 1. \) \( 12 \), this means: \( 1\cdot10^{12} \), or: \( 1,000,000,000,000 \). Other models will give similar messages once their screen capacities are surpassed; some may use a format like: \( 1 \) \( \text{E}12 \).

For these practice problems, learn to use your calculator to compute exponents and square roots. Look for keys with symbols like these: \( \wedge,\;y^{x},\;\sqrt{x} \).

Practice questions

Simplify the following

1. \( 4^{3} \)
2. \( 4^{-3} \)
3. \( 3^{5} \)
4. \( 7^{-3} \)
5. \( 10^{6} \)
6. \( 10^{-2} \)
7. \( (-2)^{3} \)
8. \( -3^{3} \)
9. \( (-3)^{-4} \)
10. \( \sqrt{64} \)
11. \( \sqrt{16} \)
12. \( \sqrt{5^{2}} \)

Convert the following numbers into scientific notation

13. \( 1,002,000,000 \)
14. \( 12,034,500,000,000 \)
15. \( 0.0000088 \)
16. \( 0.0011 \)

Convert the following numbers into standard notation

17. \( 5.6092\cdot10^{8} \)
18. \( 1.0011\cdot10^{4} \)
19. \( 3.79\cdot10^{-6} \)
20. \( 6.1602\cdot10^{-10} \)

Solutions

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1. \( 4^{3}=4\cdot4\cdot4=64 \)
2. \( 4^{-3}=\frac{1}{4^{3}}=\frac{1}{64} \)
3. \( 3^{5}=3\cdot3\cdot3\cdot3\cdot3=243 \)
4. \( 7^{-3}=1/7^{3}=1/7/7/7=1/343=\frac{1}{343} \)
5. The answer is: \( 10^{6}=1,000,000 \). Note that with \( 10^{n} \) the answer will be a \( 1 \) followed by \( n \) zeros. You can also visualize this by starting with \( 1.0 \) then moving the decimal place \( n \) spaces to the right.
6. The answer is: \( 10^{-2}=0.01 \). For solving the general case of \( 10^{-n} \), start with \( 1.0 \) and move the decimal place \( n \) spaces to the left.
7. \( (-2)^{3}=(-2)\cdot(-2)\cdot(-2)=-8 \)
8. \( -3^{3}=-1\cdot3\cdot3\cdot3=-27 \)
9. \( (-3)^{-4}=\frac{1}{(-3)^{4}}=\frac{1}{(-3)\cdot(-3)\cdot(-3)\cdot(-3)}=\frac{1}{81} \)
10. \( \sqrt{64}=8 \) since \( 8^{2}=64 \)
11. \( \sqrt{16}=4 \) since \( 4^{2}=16 \)
12. \( \sqrt{5^{2}}=5 \)
13. \( 1.002\cdot10^{9} \)
14. \( 1.20345\cdot10^{13} \)
15. \( 8.8\cdot10^{-6} \)
16. \( 1.1\cdot10^{-3} \)
17. \( 560,920,000 \)
18. \( 10,011 \)
19. \( 0.00000379 \)
20. \( 0.00000000061602 \)

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