Powers of two: learn and practice

Who this lesson is for

This powers lesson is for grades 5–6. The catalog exercises are powers of two, such as 2⁵, not a general exponent rule for every base. The practice link opens that skill at grade 5. The grade 5 and grade 6 hubs link to this one page.

The exponent counts factors

2⁵ means five factors of 2 multiplied together: 2 × 2 × 2 × 2 × 2. The 2 is the base. The 5 is the exponent. The exponent does not say to multiply 2 by 5. That product is 10, and the power is 32. Reading the small number as a factor count, rather than as something to multiply once, is the whole idea.

Build the power one factor at a time

Start at 1 and multiply by 2 once for each factor the exponent names. One factor gives 2. Two factors give 4. Three give 8. Four give 16. Five give 32. Each new factor doubles the previous power. If you can say the list 2, 4, 8, 16, 32, you can evaluate the powers this page practices.

Worked example 1: evaluate 2⁵

Evaluate 2⁵. Write five factors: 2 × 2 × 2 × 2 × 2. Pair them if you like: 2 × 2 = 4, and another 2 × 2 = 4, and one 2 is left. Then 4 × 4 = 16, and 16 × 2 = 32. So 2⁵ = 32. The wrong reading 2 × 5 = 10 uses the exponent as an ordinary factor. The small 5 counts copies of the base. It is not itself a factor in the product.

Worked example 2: evaluate 2³

Evaluate 2³. Three factors of 2 are 2 × 2 × 2. First 2 × 2 = 4, then 4 × 2 = 8. So 2³ = 8. You can also take the previous power, 2² = 4, and double it, because one more factor of 2 doubles the value. 4 × 2 = 8. Both paths agree. 2 + 2 + 2 = 6 is addition, and this notation is multiplication.

Worked example 3: evaluate 2⁶

Evaluate 2⁶. Six factors are one more than 2⁵. From example 1, 2⁵ = 32, and one more factor of 2 doubles it: 32 × 2 = 64. So 2⁶ = 64. Writing all six factors also works: 2 × 2 × 2 × 2 × 2 × 2. The doubling path is shorter once you trust the previous power. Check the count of factors. Stopping at five factors leaves you at 32, which is 2⁵, not 2⁶.

Checks before you leave the problem

Count the factors before you multiply. The exponent is that count, and the base is what you repeat. Doubling the previous power of two is a fair shortcut only when the exponent grew by 1. Then practice powers of two on the board.

Common questions about powers of two

Is 2⁵ the same as 2 × 5?

No. 2 × 5 is 10. 2⁵ is five factors of 2, which is 32. The exponent counts copies of the base.

Why only powers of two?

Those are the exponent exercises the practice catalog offers in grades 5–6. A general power rule for other bases is a later skill and is not this page.

Does 2¹ equal 2?

Yes. One factor of 2 is just 2. The exponent 1 means the base appears once, so there is nothing to multiply it by.