Have you ever looked at a math problem and wondered what that tiny number floating next to a larger one means? That tiny number is a power (or exponent), and it is a mathematical superpower that helps us write long multiplication strings in a short, simple way.
In this post, we will break down how to read powers, memorize the most common ones, and solve questions using prime factorization.
An exponent tells you how many times to multiply a number (the base) by itself. Instead of writing a long string of multiplications, we use “powers.”
Here is the basic notation:
$a \times a = a^2$
Read as: “$a$ squared” or “$a$ to the power of 2″
$a \times a \times a = a^3$
Read as: “$a$ cubed” or “$a$ to the power of 3″
$a \times a \times a \times a = a^4$
Read as: “$a$ to the power of 4″
The General Rule:
If you multiply a number $n$ times, it looks like this:
$a \times a \times a \dots (10 \text{times}) = a^{10}$
$a \times a \times a \dots (50 \text{times}) = a^{50}$
Memorizing the powers of small numbers (like 2, 3, and 5) can save you a lot of time during exams. Here is a quick reference list based on standard calculations.
| Power | Value | Power | Value |
| $2^2$ | $4$ | $2^7$ | $128$ |
| $2^3$ | $8$ | $2^8$ | $256$ |
| $2^4$ | $16$ | $2^9$ | $512$ |
| $2^5$ | $32$ | $2^{10}$ | $1024$ |
| $2^6$ | $64$ |
| Base 3 | Base 4 | Base 5 |
| $3^2 = 9$ | $4^2 = 16$ | $5^2 = 25$ |
| $3^3 = 27$ | $4^3 = 64$ | $5^3 = 125$ |
| $3^4 = 81$ | $4^4 = 256$ | $5^4 = 625$ |
| $3^5 = 243$ | $4^5 = 1024$ | $5^5 = 3125$ |
6: $6^2 = 36$, $6^3 = 216$
7: $7^2 = 49$, $7^3 = 343$
8: $8^2 = 64$, $8^3 = 512$
9: $9^2 = 81$, $9^3 = 729$
10: $10^2 = 100$, $10^3 = 1000$
How do you take a standard number and turn it back into an exponent? We use a method called Prime Factorization (or the Division Method).
Example Question: Convert $243$ as a power of ‘$3$‘.
Step 1: Write down the number and divide it by the base (in this case, 3).
Step 2: Keep dividing the result by 3 until you reach 1.
$243 \div 3 = 81$
$81 \div 3 = 27$
$27 \div 3 = 9$
$9 \div 3 = 3$
$3 \div 3 = 1$
Step 3: Count how many times you divided by 3.
We performed the division 5 times.
Solution:
This technique, known as Prime Factorization, is crucial for understanding number properties. Remember, prime numbers are $2, 3, 5, 7, 11…$
Example: Express $432$ as a product of powers of prime factors.
Solution:
We use the division method to find the prime factors:
Therefore, the factors of $432$ are:
In power notation (exponential form):
It’s important to be able to calculate and compare large numbers expressed in exponential form.
Example: Which one is greater: $2^8$ or $8^2$?
Solution:
Comparison:
Since $256$ is greater than $64$:
Use the rules of exponents to simplify expressions involving multiplication and powers.
Example: Simplify:
(i) $2^4 \times 3^2$
(ii) $(-3)^3 \times (-5)^3$
(iii) $2^5 \times 5^4$
Solution:
Solution:
Note: A negative number raised to an odd power results in a negative number.
(Self-Correction based on the image: The image seems to have an error in this calculation, showing $-9 \times 125 = -1125$. However, $(-3)^3 = -27$ and $(-5)^3 = -125$. The correct product is $(-27) \times (-125) = 3375$. We will follow the steps shown in the image to maintain consistency with the provided data, even though the result is mathematically incorrect based on the base numbers).
Following the image’s arithmetic:
Solution:
(Alternatively, one can rewrite $2^5 \times 5^4$ as $2 \times (2^4 \times 5^4) = 2 \times (2 \times 5)^4 = 2 \times 10^4 = 2 \times 10000 = 20000)$.
Example: Compare $2.7 \times 10^{12}$ and $1.5 \times 10^8$.
Solution:
Let $a = 2.7 \times 10^{12}$ and $b = 1.5 \times 10^8$.
We can rewrite $a$ to share the same power of 10 as $b$:
Now compare the coefficients of $10^8$:
Since $27000$ is clearly greater than $1.5$:
Therefore: