Lesson 73: Exponents

New Concept

Exponents are a shorthand for repeated multiplication. The exponent shows how many times the base is used as a factor.

The exponent indicates how many times the base is used as a factor.

$$7^3 \text{ means } 7 \cdot 7 \cdot 7$$

$$7^4 \text{ means } 7 \cdot 7 \cdot 7 \cdot 7$$

What’s next

This is just the beginning of working with powers. Next, you'll tackle worked examples on comparing powers and using them in prime factorization and order of operations.

Property

An exponent indicates how many times the base is used as a factor. For example, $5^3$ means $5 \cdot 5 \cdot 5$. We read expressions with exponents as powers. We say $5^2$ as "five squared" and $10^3$ as "ten cubed."

Examples

• $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$
• Compare $5^2$ and $2^5$: $5^2 = 25$ while $2^5 = 32$, so $5^2 < 2^5$.
• To simplify $10^2 + 5^2$, calculate the powers first: $100 + 25 = 125$.

Explanation

Think of an exponent as a tiny boss telling the big base number how many times to multiply itself. So, $4^3$ isn't a lazy $4 \cdot 3$, but an exciting team-up of three fours multiplying together ($4 \cdot 4 \cdot 4$). It’s a super handy shortcut for writing huge multiplication problems and making numbers grow incredibly fast!

Property
To write the prime factorization of a number, use exponents to group repeated factors. For example, the prime factorization of 1000 is $2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5$, which we can write as $2^3 \cdot 5^3$.

Examples

The prime factorization of 72 is $2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$, which is written as $2^3 \cdot 3^2$.
The prime factorization of 200 is $2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$, which is written as $2^3 \cdot 5^2$.
The prime factorization of 98 is $2 \cdot 7 \cdot 7$, which is written as $2 \cdot 7^2$.

Explanation

Instead of writing a long, messy list of prime factors, exponents let us clean things up! It’s like packing for a trip; you group all your socks together. If you have three 2s in your factorization, just write $2^3$. This bundles identical prime factors into a neat, powerful package, making the number’s secret recipe much easier to read.