Powers of Numbers

What is a Power?

If you take two natural numbers $ a $ and $ m $ with $ m > 1 $, a power with base $ a $ and exponent $ m $ is expressed as $ a^m $ and represents the product of $ m $ factors equal to $ a $: $$ a^m = a \cdot a \cdot a \cdot \ldots \cdot a \quad (\text{m times}) $$

Imagine you need to multiply the same number several times, such as $ 5 \cdot 5 \cdot 5 $. Instead of writing out this product in full, you can use a more compact notation: $ 5^3 $. In this notation:

5 is the base, the number being multiplied by itself.
3 is the exponent, indicating how many times the base is multiplied.

Using powers allows you to express the product of multiple identical factors in a concise way.

This notation is particularly useful because it simplifies calculations and makes it easier to represent very large or very small numbers.

For example, you can write the number 16 as a power of two: $$ 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16 $$ This makes calculations clearer and more efficient.

Properties of Powers
Powers with Exponent 1 and 0
Powers with Negative Exponents

Properties of Powers

Powers follow several rules that help simplify calculations. Let’s go through each of them with clear explanations and practical examples.

Product of powers with the same base
The product of two powers with the same base is a power with that base and an exponent equal to the sum of the exponents. $$ a^m \cdot a^n = a^{m+n} $$
Example: $ 2^2 \cdot 2^4 = 2^{2+4} = 2^6 = 64 $
Quotient of powers with the same base
The quotient of two powers with the same base is a power with that base and an exponent equal to the difference of the exponents, as long as the exponent of the numerator is greater than or equal to that of the denominator. $$ \frac{a^m}{a^n} = a^{m-n} \quad \text{where} \; a \neq 0, \; $$
Example: $ \frac{2^{10}}{2^7} = 2^{10-7} = 2^3 = 8 $
Product of powers with the same exponent
When multiplying two powers with the same exponent, the result is a power that keeps the same exponent and has a base equal to the product of the original bases. $$ a^n \cdot b^n = (a \cdot b)^n $$
Example: $ 5^2 \cdot 3^2 = (5 \cdot 3)^2 = (15)^2 = 225 $
Quotient of powers with the same exponent
When dividing two powers with the same exponent, the result is a power that keeps the same exponent and has a base equal to the quotient of the original bases. $$ \frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n \quad \text{where} \; b \neq 0. \; $$
Example: $ \frac{15^{2}}{3^2} = \left( \frac{15}{3} \right)^2 = 5^2 = 25 $
Power of a power
The power of a power results in a power with the same base and an exponent equal to the product of the exponents. $$ (a^m)^n = a^{m \cdot n} $$
Example: $ (2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64 $
Power of a product
The power of a product equals the product of the powers of each factor. $$ (a \cdot b)^m = a^m \cdot b^m $$
Example: $ (3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = 144 $
Power of a quotient
The power of a quotient is the quotient of the powers of the numerator and the denominator, as long as the denominator is not zero. $$ \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \quad \text{where} \; b \neq 0 $$
Example: $ \left(\frac{12}{4}\right)^2 = \frac{12^2}{4^2} = \frac{144}{16} = 9 $

These properties are essential for simplifying complex expressions and solving problems involving powers quickly and efficiently.

Practical Examples

When you multiply two powers with the same base, such as $ 2^2 $ and $ 2^4 $, you add the exponents:

$$ 2^2 \cdot 2^4 = 2^{2+4} = 2^6 = 64 $$

When you divide two powers with the same base, you subtract the exponents:

$$ \frac{2^{10}}{2^7} = 2^{10-7} = 2^3 = 8 $$

For a power raised to another exponent, you multiply the exponents together:

$$ (2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64 $$

When raising a product, each factor is raised separately:

$$ (3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = 144 $$

Similarly, the power of a quotient is found by raising both the numerator and denominator separately:

$$ \left(\frac{12}{4}\right)^2 = \frac{12^2}{4^2} = \frac{144}{16} = 9 $$

These are just a few practical examples to help you get familiar with using powers.

Why Do Powers Simplify Calculations? Consider the expression: $$ \frac{(16^4 \cdot 16^9)^2}{16^{29}} $$ Performing all the calculations directly would take time, but using the properties of powers simplifies it quickly with basic arithmetic. For example, when multiplying powers with the same base, you add the exponents: $$ \frac{(16^4 \cdot 16^9)^2}{16^{29}} = \frac{(16^{4+9})^2}{16^{29}} = \frac{(16^{15})^2}{16^{29}} $$ The power of a power is simplified by multiplying the exponents: $$ \frac{(16^{15})^2}{16^{29}} = \frac{16^{15 \cdot 2}}{16^{29}} = \frac{16^{30}}{16^{29}} $$ Lastly, when dividing powers with the same base, you subtract the exponents: $$ \frac{16^{30}}{16^{29}} = 16^{30-29}= 16 $$ The final result is 16. So, by using the properties of powers, the expression simplifies efficiently without lengthy calculations.

Powers with Exponent 1 and 0

The definition of a power applies when the exponent is greater than 1, as it makes sense to talk about multiplication only with at least two factors.

However, it’s also useful to define what happens when the exponent is 1 or 0:

Exponent 1
If the exponent is 1, the power of a number $ a $ is simply $ a $ itself: $$ a^1 =a $$
By definition, $ a^1 = a $. This is intuitive: if you multiply a number by itself once, the result is the number itself.
For example, if you have the number 7 and raise it to the power of 1, $ 7^1 $, it equals 7: $$ 7^1 = 7 $$ This makes sense since raising a number to the power of 1 is the same as multiplying it by itself only once, leaving the original number unchanged.
Exponent 0
For any natural number $ a $ other than 0, it’s defined that $ a^0 = 1 $. $$ a^0 =1 $$
This might seem less intuitive, but it aligns with mathematical rules. For example, considering the property $ a^m \cdot a^n = a^{m+n} $, it’s clear that for this rule to hold when $ n = 0 $, we must have $ a^0 = 1 $. $$ a^n \cdot a^0 = a^{n+0} = a^n $$ Another way to demonstrate that $ a^0 = 1 $ is by applying the rule for dividing powers: $$ \frac{a^n}{a^n} = a^{n-n} = a^0 $$ The left side simplifies to 1, so $ a^0 $ must be 1 to maintain consistency.
For instance, if you multiply $ 2^3 $ by $ 2^0 $, using the property of powers: $$ 2^3 \cdot 2^0 = 2^{3+0} $$ $$ 2^3 \cdot 2^0 = 2^3 $$ The value remains $ 2^3 $. This shows that multiplying by $ 2^0 $ doesn’t change the result, implying $ 2^0 $ must be 1: $$ 2^3 \cdot 1 = 2^3 $$ Clearly, $ 2^0 = 1 $.

A special case is $ 0^0 $, which is left undefined because there is no consensus on its value. Some leave it undefined to avoid ambiguity.

Why is 0⁰ Undefined?

The ambiguity of $ 0^0 $ comes from two conflicting interpretations in algebra:

According to $a^0 = 1$, $0^0$ should be $1$.
According to $0^n = 0$, $0^0$ should be $0$.

These two rules lead to opposing conclusions, making $ 0^0 $ both valid and contradictory depending on the approach.

To avoid ambiguity, $0^0$ is considered undefined.

This choice prevents contradictions and inconsistencies.

Another way to demonstrate that $ 0^0 $ is undefined is to write it as $ 0^{n-n} $ with $ n > 0 $. $$ 0^0 = 0^{n-n} $$ Applying power rules, this becomes a division: $$0^0 = 0^{n-n} = \frac{0^n}{0^n} $$ Since $ 0^n = 0 $ when $ n > 0 $, this division leads to a division by zero, which is undefined in mathematics. Therefore, $ 0^0 $ must also be undefined.

Powers with Negative Exponents

Powers with negative exponents are introduced only when working with rational numbers.

For any base $ a \neq 0 $ and any natural number $ n $, a power with a negative exponent is defined as the reciprocal of the corresponding power with a positive exponent: $$ a^{-n} = \frac{1}{a^n} $$

It is crucial that the base is nonzero $ a \neq 0 $, as the reciprocal of zero is undefined. Therefore, powers with negative exponents do not exist if the base is zero.

In general, a negative exponent indicates the reciprocal of a power with a positive exponent.

For instance, raising 5 to the power of -1 gives the reciprocal of 5:

$$ 5^{-1} = \frac{1}{5^1} = \frac{1}{5} $$

Similarly, raising 3 to the power of -2 gives the reciprocal of 3 squared:

$$ (3)^{-2} = \frac{1}{(3)^2} = \frac{1}{9} $$

Pay close attention to the sign: if the base is negative, the sign of the result depends on the exponent. If the exponent is even, the result is positive; if it is odd, the result is negative.

$$ (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9} $$

If the base is a fraction, such as $ \frac{3}{2} $, the process is exactly the same. Simply treat the base as $ a = \frac{3}{2} $:

$$ \left( \frac{3}{2} \right)^{-2} = \left( \frac{1}{\frac{3}{2}} \right)^2 = \left( \frac{2}{3} \right)^2 = \frac{4}{9} $$

By extending the concept of exponents, we can assign meaning to symbols like $ 2^{-3} $, while preserving the fundamental properties of exponents, such as:

$$ a^m \cdot a^n = a^{m+n} \quad \text{even for negative exponents} $$

Although it may seem abstract at first, this concept becomes intuitive with the right explanation.

Take the example $ 2^{-3} $. To preserve the rules of exponents, we know:

$$ 2^{-3} \cdot 2^3 = 2^{-3 + 3} = 2^0 = 1 $$

For this equation to hold true, $ 2^{-3} $ must be the reciprocal of $ 2^3 $. This leads to the fundamental definition:

$$ 2^{-3} = \frac{1}{2^3} $$

Keep in mind that the result of a power with a negative exponent is not always negative or positive. A negative exponent does not automatically mean a negative result. The sign of the result depends entirely on the sign of the base and whether the exponent is even or odd. It has nothing to do with the sign of the exponent itself. So, don’t confuse the sign of the base with the sign of the exponent. For example, in $ (-2)^{-3} $, the result will be negative because the base is negative and the exponent is odd: $$ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8} $$ Conversely, in $ (-2)^{-2} $, the result will be positive because the base is negative and the exponent is even: $$ (-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4} $$