Base and exponent

The idea

nᵃ denotes n multiplied by itself a times, read 'n raised to the power a', where n is the base and a is the exponent or power; a × a × a × b × b is written a³b².

Seven twos is a lot to write

When the paper was folded 7 times, its thickness was 0.001 cm × 2 × 2 × 2 × 2 × 2 × 2 × 2. Mathematicians write it more shortly: 0.001 cm × 2^7.

What do you think the small 7 in 2^7 tells us?

• Multiply 2 by 7
• Write 2 seven times and multiply them
• Add 2 seven times

Count the factors

Change the base and the exponent. Count the tiles each time.

Now with letters. Tap to build a product and watch the short form.

The exponent counts

n^a means n multiplied by itself a times. It is read “n raised to the power a”. n is the base, and a is the exponent (or power).

5^4 = 5 × 5 × 5 × 5 = 625. (−4)^3 = (−4) × (−4) × (−4) = −64. a × a × a × b × b = a^3b^2.

Careful: 4 + 4 + 4 = 3 × 4 = 12, but 4 × 4 × 4 = 4^3 = 64.

Notes

n^a means n multiplied by itself a times: n is the base, a is the exponent (power). a × a × a × b × b = a^3b^2.

Check yourself

What does 4^3 mean?

What is 2^5?

Answer: 32

2 × 2 × 2 × 2 × 2 = 32.

In 7^4, which is the base and which is the exponent?

32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3 Write it with exponents.

• 4 × 3 = 12. That would be 4 + 4 + 4. A power multiplies the base by itself.
• 4 × 4 × 4 = 64 — correct. Yes! The base 4 is multiplied by itself 3 times.
• 3 × 3 × 3 × 3 = 81. You swapped them. The base (4) is what is multiplied; the exponent (3) says how many times.
• Base 7, exponent 4 — correct. Yes! 7 is multiplied by itself; 4 counts the 7s.
• Base 4, exponent 7. The other way round. The small raised number is the exponent.
• Both are bases. Only the big number is the base. The small raised number is the exponent.
• 2^4 × 5^2 × 3^4 — correct. Yes! Four 2s, two 5s, four 3s.
• 2^4 × 5^4 × 3^2. Count again: there are two 5s and four 3s.
• 4^2 × 2^5 × 4^3. The bases must be the repeated numbers 2, 5 and 3, and the exponents count them.