What Exponents Actually Mean
The Meaning of Exponents
- Multiplication is repeated addition. For example, \(3\cdot 4\) literally means \(4+4+4\), or “three 4’s added together.”
- Similarly, \(5x=x+x+x+x+x\), or “five x’s added together”. This idea will be helpful later when we talk about polynomials.
- Similar to multiplication, exponentiation is repeated multiplication.
- For example, \(3^4=3\cdot 3\cdot 3\cdot 3\), or “3 multiplied by itself 4 times.”
Thinking in these terms, i.e. in terms of what a specific notation means, will be useful throughout this and every math class you take… and in life too!
Exponents of Negative Numbers
You probably remember from “way back when” that when you multiply two negative numbers, you get a positive number. This idea, along with the fact that exponents mean “repeated multiplication,” we get the following principle.
Compute \((-2)^6\) and \((-2)^7\)
Solution
Note that for \((-2)^6\), we can expand the power so that we have repeated multiplication of the \(-2\), six times.
$$(-2)^6=(-2)(-2)(-2)(-2)(-2)(-2)$$
Then, because we know that negative times negative is positive, pairing off \((-2)\)’s and multiplying each pair gives us
$$\begin{align}(-2)(-2)(-2)(-2)(-2)(-2) &=((-2)(-2))\cdot((-2)(-2))\cdot ((-2)(-2))\\&=(4)\cdot(4)\cdot(4)\\&=64\end{align}$$
The same idea could applied for any even power, such as \(2\) or \(300\); just pair off the factors and multiply the two numbers in each pair to get positive numbers. The product of positive numbers is positive, so the end result will always be positive.
Now, note for \((-2)^7\), we can expand the power so we get
$$(-2)^7=(-2)(-2)(-2)(-2)(-2)(-2)(-2)$$
However, since we had an odd power, we will not be able to pair off all \((-2)’s\) such that all the negatives cancel out; there is one extra \((-2)\) because of this odd power. All but this spare \((-2)\) will be part of a “cancelling” pair of negatives. Thus, after multiplying out the individual pairs, we get a big positive times a \((-2)\). The following shows this in action
$$\begin{align}(-2)(-2)(-2)(-2)(-2)(-2)(-2) &= ((-2)(-2))\cdot ((-2)(-2))\cdot((-2)(-2))\cdot(-2) \\&=(4)\cdot(4)\cdot(4)\cdot(-2)\\&=64\cdot (-2)\\&=-128\end{align}$$
This same exact reasoning could be applied no matter what odd power we are dealing with, hence why the principle stated above is true.
Naturally, you don’t need to do the whole “pair and cancel” thing shown above; which was there to illustrate the aforementioned fundamental principle of powers of negatives.
As an easy shortcut, when dealing with even powers of negatives such as \((-5)^4\), since we know that all the negatives “pair off” and cancel, we can simply ignore the minus sign, and just compute \(5^4\), which gives the final answer of \(625\).
Same thing can be done for single variables representing unknown quantities that have a negative out front. For example:
$$\begin{align} (-x)^{10}&=x^{10}\end{align}$$
Now, when dealing with odd powers of negatives such as \((-5)^3\), because all but one \((-5)\) gets paired off, cancelling all but one negative sign (as per the above explanation), the minus sign sticks around until the end. So, we can compute \(5^3\) first, then tack on a minus sign at then end.
Same thing can be done for single variables representing unknown quantities with a minus sign out in front:
$$\begin{align} (-x)^{7}=-(x)^7=-x^7\end{align}$$
In short: minus signs inside even powers vanish/cancels out. Minus signs inside odd powers stick around (in the final answer).
Examples of exponential expressions are:
| $$5\cdot 2^7 $$ | $$3x^2$$ | $$\frac{1}{2}y^{2.81} $$ |
| $$6.712(1.08)^5 $$ | $$ x^2 $$ | $$xy^{x} $$ |
Note that an exponential expression does NOT need to have a (listed) coefficient, such as in the case of \(x^2\). Here the coefficient is technically \(1\), but we don’t bother writing it, because multiplying by \(1\) doesn’t do anything.
In short, an exponential expression is an expression that is “something raised to some power, with a number (maybe) out in front.“
A Quick Note (that will save you a lot of lost points)
The exponent in an exponential expression only applies to the quantity to which it is attached.
For example \((2x)^3\) is NOT the same as \(2x^3\).
In \((2\cdot 5)^3\), the power \(3\) is attached to the whole quantity \((2\cdot 5)\), so when computing, you get \((2\cdot 5)^3=(10)^3=1000\). However, in \(2\cdot 5^3\) the power \(3\) is only attached to the \(5\) and NOT the \(2\), and thus only applies to the \(5\): \(2\cdot 5^3=2\cdot (125)=250\).
So, if you want to include a number or variable inside a power, put everything that needs to be raised to that power inside parentheses.
Similarly, when you want to raise a negative to a power, you must include the negative inside parentheses to prevent ambiguity. For instance \((-2)^6\) is NOT equal to \(-2^6\); the power \(6\) in \(-2^6\) only applies to the number \(2\), and not the negative in front of it.
\(2^4=16\)
\(\left(\frac{1}{2}\right)^4=\frac{1}{16}\)
One way you can compute these by hand is by multiplying values together sequentially one at a time:
\(2^4=2\cdot 2\cdot 2\cdot 2=4\cdot 2\cdot 2=8\cdot 2=16\)
\( \left(\frac{1}{2} \right)^4= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} \cdot \frac{1}{2} = \frac{1}{16} \)
\(3^4=81\)
\(\left(\frac{2}{3}\right)^4=\frac{16}{81}\)
One way you can compute these by hand is by multiplying values together sequentially one at a time:
\(3^4=3\cdot 3\cdot 3\cdot 3=9\cdot 3\cdot 3=27\cdot 3=81\)
\( \left(\frac{2}{3} \right)^4= \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{8}{27} \cdot \frac{2}{3} = \frac{16}{81} \)
\((-3)^4=81\)
\(-3^4=-81\)
\(-3^5=-243\)
\((-3)^5=-243\)
In the case of \((-3)^4\) and \((-3)^5\), the exponent applies to everything inside the parentheses, including the minus sign.
So for \((-3)^4\), using the shortcut mentioned in the Key Takeaways, since we have an even power of a negative, the negative gets cancelled, leaving us with the computation \(3^4=81\).
For \((-3)^5\), the power is odd, so we compute \(3^5=243\) and put a minus sign in front of it (because the power cancels out all but one minus sign).
Note that in \(-3^4\) and \(-3^5\) the powers only apply to the \(3\) and NOT the negative out in front. So the negative comes along for the ride to the final answer while you compute \(3^4\) and \(3^5\).