Evaluating a Power

Calculate $3^4$

Identify base and exponent: Base = 3, Exponent = 4 = Multiply 3 by itself 4 times

Write out the multiplication: $3^4 = 3 \times 3 \times 3 \times 3$ = Four 3s multiplied

Multiply step by step: $3 \times 3 = 9$, then $9 \times 3 = 27$, then $27 \times 3 = 81$ = 81

Answer: $3^4 = 81$

Converting to Exponential Form

Write $5 \times 5 \times 5 \times 5 \times 5 \times 5$ using an exponent.

Identify the base: The number being repeated is 5 = Base = 5

Count the factors: Count how many 5s: six 5s multiplied together = Exponent = 6

Write in exponential form: $5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6$ = $5^6$

Answer: $5^6$ (which equals 15,625)

Powers of 10

What is $10^6$?

Recognize the pattern: Powers of 10 create numbers with zeros = The exponent tells us how many zeros

Apply the rule: $10^6$ means 1 followed by 6 zeros = 1,000,000

Name the number: 1,000,000 is one million = One million

Answer: $10^6 = 1,000,000$ (one million)

Comparing Powers

Which is greater: $2^8$ or $8^2$?

Calculate $2^8$: $2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256$ = 256

Calculate $8^2$: $8^2 = 8 \times 8 = 64$ = 64

Compare: $256 > 64$ = $2^8 > 8^2$

Answer: $2^8 = 256$ is greater than $8^2 = 64$

Mistake: Multiplying base by exponent: $3^4 = 12$

Why: This confuses exponentiation with multiplication. The exponent tells you how many times to multiply, not what to add.

Correct: $3^4 = 3 \times 3 \times 3 \times 3 = 81$, not $3 \times 4 = 12$

Mistake: Thinking $2^3 = 2^2 + 2^1$

Why: Exponents don't work like regular addition. You can't split them this way.

Correct: $2^3 = 8$, while $2^2 + 2^1 = 4 + 2 = 6$. These are different!

Mistake: Writing $5^2$ as $25^1$ and thinking they're different

Why: Any number to the power of 1 equals itself: $25^1 = 25$

Correct: $5^2 = 25$ and $25^1 = 25$ are the same value, just written differently

Computer Memory

Computer memory is measured in powers of 2 because computers use binary (0s and 1s).

1 Kilobyte = $2^{10}$ = 1,024 bytes. 1 Megabyte = $2^{20}$ = 1,048,576 bytes.

Bacterial Growth

Bacteria multiply by splitting in two. Each hour, the population doubles.

Starting with 1 bacterium, after 10 hours you have $2^{10}$ = 1,024 bacteria!

Area and Volume

Exponents naturally appear in geometry: area uses squared, volume uses cubed.

A square with side 7 cm has area $7^2 = 49$ square cm. A cube with side 4 cm has volume $4^3 = 64$ cubic cm.

Q: What is any number to the power of 0?

A: Any non-zero number to the power of 0 equals 1. For example, $5^0 = 1$, $100^0 = 1$. This will be covered in detail in a future lesson!

Q: Why is it called 'squared' and 'cubed'?

A: 'Squared' comes from calculating the area of a square (side times side). 'Cubed' comes from calculating the volume of a cube (side times side times side).

Q: Is $2^3$ the same as $3^2$?

A: No! $2^3 = 8$ (two multiplied three times) but $3^2 = 9$ (three multiplied two times). The order matters!