Powers of 10 & Whole-Number Ops

5.B.1Multiply / divide by 10, 100, 100001/04

Multiply and divide whole numbers by 10, 100, 1000.

Step01of 04

Multiplying by 10 SHIFTS digits one place to the LEFT (and you append a zero, since the ones place becomes empty).

37 × 10	→370
37 × 100	→3,700
37 × 1,000	→37,000

5.B.1 — Multiply and divide whole numbers by 10, 100, 1000.

Step 01 of 04

Multiplying by 10 SHIFTS digits one place to the LEFT (and you append a zero, since the ones place becomes empty).

37 × 10	→370
37 × 100	→3,700
37 × 1,000	→37,000

Step 02 of 04

Dividing by 10 shifts digits one place to the RIGHT.

400 ÷ 10	→40
400 ÷ 100	→4
4,000 ÷ 1,000	→4

Step 03 of 04

Why? Each shift left multiplies by 10 (each place is worth 10× the place to its right). Same logic in reverse for division.

Step 04 of 04

Word problem. A box holds 24 cans. How many cans in 100 boxes?

24 \times 100 = 2, 400

Key insight

Multiply by 10ⁿ → shift digits LEFT n places. Divide by 10ⁿ → shift RIGHT n places. No actual multiplication / division needed.

5.B.2 — Powers of 10 with exponents.

Step 01 of 04

Exponent notation. $1 0^{n}$ means "10 multiplied by itself n times."

$1 0^{1}$	→10
$1 0^{2}$	→100
$1 0^{3}$	→1,000
$1 0^{4}$	→10,000
$1 0^{5}$	→100,000
$1 0^{6}$	→1,000,000

Step 02 of 04

The EXPONENT equals the number of zeros after the 1.

1 0^{7} = 10, 000, 000

Step 03 of 04

Multiplying with powers of 10. Same as before — the exponent tells you how many places to shift.

8.5 \times 1 0^{3} = 8, 500

123 \times 1 0^{4} = 1, 230, 000

Step 04 of 04

Compare big numbers using powers of 10. 10⁵ is 10× as big as 10⁴. Each step up multiplies by 10.

Key insight

Exponent on 10 = number of zeros (or shift amount). Multiplying by

1 0^{n}

shifts digits n places left.

5.B.3 — Multi-digit multiplication (standard algorithm).

Step 01 of 04

Same standard algorithm as grade 4, just with bigger numbers. Multiply each digit of the bottom number by the entire top number, with placeholder zeros for tens, hundreds, etc.

Step 02 of 04

Worked example. 234 × 56.

Multiply by ones (6): $234 \times 6 = 1, 404$ .

Multiply by tens (5): $234 \times 5 = 1, 170$ , write as 11,700 (placeholder zero).

Step 03 of 04

Add the partial products.

1, 404 + 11, 700 = 13, 104

234 \times 56 = 13, 104

Step 04 of 04

Estimate first. 234 × 56 ≈ 200 × 60 = 12,000. Our answer 13,104 is close — sanity check passed.

Key insight

Multiply by each digit of the bottom number, with placeholder zeros for higher places. Add partial products. Always estimate first to catch big mistakes.

5.B.4 — Divide 4-digit numbers by 2-digit (long division).

Step 01 of 04

Same divide-multiply-subtract-bring-down rhythm as grade 4, but with 2-digit divisors. The trick is ESTIMATING how many times the divisor goes into the working number — usually requires guess-and-check.

Step 02 of 04

Worked example. 4,896 ÷ 24.

Can 24 go into 4? No. Into 48? Yes — 2 times (24 × 2 = 48). Write 2 above the 8. Subtract 48 − 48 = 0. Bring down 9.

Step 03 of 04

Continue. 24 into 09 → 0 times. Write 0 above. Bring down 6 to make 96. 24 into 96 → 4 times (24 × 4 = 96). Write 4. Subtract 0. No more digits.

4, 896 \div 24 = 204

Step 04 of 04

Estimating quotient digits. Round the divisor to the nearest 10 to guess. 24 ≈ 20, so 24 into 96 → 96 / 20 ≈ 4 — try 4. Adjust if too big or too small.

Key insight

Divide-multiply-subtract-bring down. With 2-digit divisors, estimate using the closest round number. Adjust your guess if needed.

5.B.5 — Multi-step word problems (whole numbers).

Step 01 of 04

Multi-step problems chain operations. Read carefully, identify each piece, do them in the right order.

"first ... then ..."	→sequential operations
"in total"	→final addition
"each ... how many in all"	→multiply

Step 02 of 04

Worked example. A school has 24 classrooms with 32 students each. 75 students go on a field trip. How many remain?

Step 03 of 04

Step 1 — total students:

24 \times 32 = 768

Step 2 — subtract those on the trip:

768 - 75 = 693 remain

Step 04 of 04

Order matters. Always answer the LAST question, not an intermediate one.

Key insight

Break the problem into steps. Solve one piece at a time. Reread the question to make sure you're answering what's asked, not just the first thing you can compute.

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