Powers and square roots

Multiplication is a mathematical function that, simply put, is used to write repeated multiplication in abbreviated form.

The natural exponent

Multiplication has the form aⁿ, where we call the expression n the exponent and the expression a the base. Next, we will be primarily interested in the shape of the exponent. In this section, we will show the properties of a power with a natural exponent, that is, when the exponent is a number 1, 2, 3, 4, …

An example of such a power is 6². Six is the base, two is the exponent. The exponent tells us how many times in a row to multiply six to get the result. So 6² = 6 · 6 = 36. Another example is 4³ = 4 · 4 · 4 = 64. In general, we could write it like this:

$$a^n = \underbrace{a\cdot a \cdot \ldots \cdot a}_{n\mbox{ -times }}$$

The base can be anything - a number, but feel free to use a more complex expression. Then, during the calculation, you just multiply the given expression as many times as the exponent indicates. Examples:

$$\begin{eqnarray} (-5)^3&=&(-5)\cdot(-5)\cdot(-5)\\ x^4&=&x\cdot x\cdot x\cdot x\\ (x+3)^2&=&(x+3)\cdot(x+3) \end{eqnarray}$$

For an exponent that is zero, we introduce the equality a⁰ = 1.

For a negative exponent, we use the following

In this section, we will assume that the exponent will be negative. What could be the interpretation? We assume that a⁰ = 1. If we want to compute a¹, then we could say that we multiply a⁰ by the expression a. Since a⁰ = 1, we get a⁰ · a = a by the product. We get a. If we want to compute a², we could write a² = a⁰ · a · a.

We can say that the value of a⁰ = 1 is our default value and when we calculate aⁿ, we just n-times multiply the value of a⁰ by the expression a. How would we proceed if n were negative? If n is positive, we multiply. If n is negative, we divide. So we would get a⁻¹ by taking the initial value of a⁰ and dividing that value by the expression a. This gives us the equation:

$$a^{-1}=\frac{a^0}{a}=\frac1a$$

If we wanted to know a⁻², we would divide a⁰ by the expression a twice. What would we get? First, let's see how we can write the division differently. If we have the quotient x/y, we might as well write it as

$$x\cdot\frac1y$$

This is one of the basic properties of fractions. So if we want to divide an expression x twice by an expression y, we can write it as

$$x\cdot\frac1y\cdot\frac1y$$

Let's go back to the example of a⁻². We said that we get this by dividing the expression a⁰ twice by a. We further modify this as follows:

$$a^{-2}=a^0\cdot\frac{1}{a}\cdot\frac1a=a^0\frac{1}{a\cdot a}=1\cdot\frac{1}{a^2}=\frac{1}{a^2}$$

At this point, we have a procedure to calculate a power with a negative exponent. We calculate the power as if the exponent were positive, and then just invert the value by dividing one slash by the result of the exponent. Written more precisely:

$$a^{-n}=\frac{1}{a^n}$$

(Here we assume that −n is a negative number, so n is positive.)

So a few examples:

$$\begin{eqnarray} 2^{-1}&=&\frac{1}{2^1}=\frac12\\ 5^{-3}&=&\frac{1}{5^3}=\frac{1}{125}\\ (2x+3)^{-8}&=&\frac{1}{(2x+3)^8} \end{eqnarray}$$

Rational exponent

We can further expand the exponent for all rational numbers. A rational number is a number that can be expressed as a fraction, the quotient of two integers. So let's have a fraction of the form m/n, where n is a positive number. Then we can write the formula

$$\Large a^{\frac{m}{n}}=\sqrt[n]{a^m}$$

The squiggle above a^m is the sign for the square root.

Calculator

If you need to calculate a power, you can use the power calculator here 🧮.

Properties of powers

• a⁰ = 1If a ≠ 0.
• a¹ = a.
• 0ⁿ = 0, if n > 0.
• 0⁰ is an undefined expression.
• (a · b)ⁿ = aⁿ · bⁿ.
• a^m · aⁿ = a^{m + n}.
• $\Large \frac{a^m}{a^n}=a^{m-n}$, if a ≠ 0.
• (a^m)ⁿ = a^{m · n}.