SME
Grade 8 Math · Problem type
Multiply in scientific notation
Use this for “(3.1 × 10⁴) × (2 × 10⁶)”.
Diagram
62,000,000,000 = 6.2 × 10¹⁰
10¹⁰ ≤ 62,000,000,000 < 10¹¹
This is an example. Type your own values to start a new problem.
(
a
×
10
)
×
(
c
×
10
)
=
p
×
10
Formulas
p
=
a
×
c
,
moved
to
be
from
1
up
to
10
(front
of
the
product
=
front
of
the
first
×
front
of
the
second,
moved
to
be
from
1
up
to
10)
6.2 = 3.1 × 2, moved to be from 1 up to 10
e
=
n
+
k
,
plus
1
if
a
×
c
reached
10
(power
of
the
product
=
power
of
the
first
+
power
of
the
second,
plus
1
if
front
of
the
first
×
front
of
the
second
reached
10)
10 = 4 + 6, plus 1 if 3.1 × 2 reached 10
Assumptions
•
Multiply the fronts and add the exponents.
•
If the front reaches 10, move the point once and add 1 to the exponent.
•
Dividing works the same way: divide the fronts and subtract the exponents.
Step-by-step
WE KNOW
a = 3.1 (front of the first), n = 4 (power of the first), c = 2 (front of the second), k = 6 (power of the second)
FIND
front of the product (p), power of the product (e)
Step 1 · Find front of the product (p)
Use p = a × c, moved to be from 1 up to 10 (front of the product = front of the first × front of the second, moved to be from 1 up to 10)
Multiply the fronts.
p
=
3.1
×
2
p
=
6.2
Step 2 · Find power of the product (e)
Use e = n + k, plus 1 if a × c reached 10 (power of the product = power of the first + power of the second, plus 1 if front of the first × front of the second reached 10)
Add the exponents.
e
=
4
+
6
e
=
10
Check
Put the numbers back in. Both sides should match.
3.1 × 2 = 6.2 ✓
4 + 6 = 10 ✓
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Add and subtract in scientific notation
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Related lessons
Scientific notation
Main lesson
Which is bigger, and how many times
Use this for “Which is greater, 2.4 × 10⁷ or 6.5 × 10⁴, and about how many times?”
Add and subtract in scientific notation
Use this for “3.6 × 10⁵ + 2.9 × 10⁵” or “How much larger is 4.2 × 10⁸ than 7.5 × 10⁷?”
Powers of ten by name
Use this for “Match 1,000,000 to one million and to a power of ten.”