Exponents

An exponent is a shortcut for repeated multiplication. In the power $a^{n}$:

• $a$ is the base --- the number being multiplied.
• $n$ is the exponent (or power) --- how many times the base is used as a factor.

We read $a^{n}$ as “$a$ to the $n$th power.” Special names: $a^{2}$ is “$a$ squared” and $a^{3}$ is “$a$ cubed.”

For any nonzero base(s) $a$, $b$ and integers $m$, $n$:

• Product of powers: $a^{m}\cdot a^{n} = a^{m+n}$
• Quotient of powers: $\dfrac{a^{m}}{a^{n}} = a^{m-n}$
• Power of a power: $\left(a^{m}\right)^{n} = a^{mn}$
• Power of a product: $\left(ab\right)^{n} = a^{n}b^{n}$
• Power of a quotient: $\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}$
• Zero exponent: $a^{0} = 1$
• Negative exponent: $a^{-n} = \dfrac{1}{a^{n}}$   and   $\dfrac{1}{a^{-n}} = a^{n}$

The whole topic comes back to one idea: same base, combine the exponents.

When you multiply powers with the same base, keep the base and add the exponents.

Why? $a^{2}\cdot a^{3} = (a\cdot a)(a\cdot a\cdot a) = a^{5}$: you are just counting all the factors.

When you divide powers with the same base, keep the base and subtract the bottom exponent from the top.

The shared factors cancel: $\dfrac{a^{5}}{a^{2}} = \dfrac{a\cdot a\cdot a\cdot a\cdot a}{a\cdot a} = a^{3}$.

When a power is raised to another power, keep the base and multiply the exponents.

Because $\left(a^{2}\right)^{3} = a^{2}\cdot a^{2}\cdot a^{2} = a^{6}$.

An exponent on a product or a quotient applies to every factor inside.

Any nonzero base raised to the $0$ power equals $1$:

This follows from the quotient rule: $\dfrac{a^{3}}{a^{3}} = a^{3-3} = a^{0}$, and any number divided by itself is $1$.

A negative exponent means take the reciprocal (flip it), then use the positive exponent:

A factor moves between numerator and denominator by flipping the sign of its exponent.

Real problems mix several rules. A reliable order:

• Apply power-of-a-power / power-of-a-product to clear outer exponents.
• Multiply (add exponents) and divide (subtract exponents) like bases.
• Rewrite any negative exponents as positive by flipping their factors.
• Simplify the number coefficients.

A fully simplified answer usually has no negative exponents and each base appearing once.

Scientific notation writes a number as

The factor $c$ (the coefficient) has exactly one nonzero digit before the decimal point. The power of $10$ records how far the decimal point moved.

• Large numbers use a positive exponent: $4\,300 = 4.3\times 10^{3}$.
• Small numbers (between $0$ and $1$) use a negative exponent: $0.0057 = 5.7\times 10^{-3}$.

Handle the coefficients and the powers of $10$ separately, using the product and quotient rules on the $10$'s.

At the end, adjust so the coefficient again satisfies $1 \le c < 10$.

Many real situations follow

where $a$ is the starting amount (the value when $x=0$) and $b$ is the growth factor. The value of $b$ decides the behavior:

• Growth: $b > 1$ --- the quantity gets larger as $x$ increases (doubling, tripling, interest).
• Decay: $0 < b < 1$ --- the quantity gets smaller as $x$ increases (halving, cooling, fading).

Because $b^{0}=1$, the starting value is always $y = a\cdot b^{0} = a$.

If two powers with the same base are equal, then their exponents must be equal:

To solve an equation like $b^{\,\square} = N$, rewrite both sides as powers of one common base, then set the exponents equal and solve that (usually linear) equation.

• Ask: “What base makes both sides simple?” Often it is the smaller of the two numbers.
• Use the power-of-a-power rule to fold the outside exponent in: $\left(b^{k}\right)^{x} = b^{kx}$.

A fractional exponent is a compact way to write a root. For $a \ge 0$ and positive integers $m,n$:

Read the fraction as “root on the bottom, power on the top”: the denominator $n$ is the index of the radical, and the numerator $m$ is the ordinary power. You may take the root first or the power first --- the answer is the same, but taking the root first keeps the numbers small.

You cannot just eyeball which of $2^{40}$ or $3^{30}$ is larger. The trick is to make either the exponents match or the bases match, then compare.

• Match the exponents: factor the exponents to pull out a common one, then compare the resulting bases.
• Match the bases: rewrite both as powers of a shared base when possible.

Once one part agrees, the larger base (or larger exponent) wins.

The units digit (last digit) of a power repeats in a short cycle as the exponent grows, because only the previous units digit affects the next one. For base $7$:

so the units digits run $7,\,9,\,3,\,1,\,7,\,9,\,3,\,1,\dots$ --- a cycle of length $4$. To find the units digit of $7^{k}$, divide $k$ by the cycle length and use the remainder to pick the spot in the cycle (a remainder of $0$ lands on the last entry).

The model $y = a\cdot b^{x}$ often comes from a percent change per step. Writing the growth factor as $b = 1 \pm r$ (with $r$ a decimal rate) separates the two cases:

Here $a$ is the starting amount and $r$ is the rate per period. Growth uses $1+r>1$; decay uses $1-r$ between $0$ and $1$.

Exponentiation asks “what do I get when I raise $b$ to the power $x$?” The logarithm asks the reverse: “what power of $b$ gives me $y$?” They are inverse operations:

So $\log_{b} y$ is simply the exponent you would put on $b$ to land on $y$. For example, $\log_{2} 8 = 3$ because $2^{3}=8$, and $\log_{10} 1000 = 3$ because $10^{3}=1000$. This is the tool that solves $2^{x}=10$, where the bases cannot be matched by hand.

For multiplying and dividing you handle the coefficients and the powers of $10$ separately. But to add or subtract, the powers of $10$ must first match. Rewrite the smaller-exponent number so both share the larger exponent, then add or subtract the coefficients:

Finally, adjust the coefficient back into the range $1 \le c < 10$ if needed.