SME
Grade 8 Math · Problem type
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?”
Diagram
650,000 = 6.5 × 10⁵
10⁵ ≤ 650,000 < 10⁶
10⁵ = 100,000
10⁶ = 1,000,000
g = 2.9 (second front at the first power)
o = + (add or subtract)
This is an example. Type your own values to start a new problem.
a
×
10
c
×
10
=
u
×
10
g
Second front at the first power
worked out (not typed)
S
Answer
worked out (not typed)
Formulas
g
=
c
×
10^(
k
−
n
)
(second
front
at
the
first
power
=
front
of
the
second
×
10^(power
of
the
second
−
power
of
the
first))
2.9 = 2.9 × 105 − 5
S
=
(
a
+
or
−
g
)
×
10^
n
,
as
o
says
(answer
=
(front
of
the
first
+
or
−
second
front
at
the
first
power)
×
10power
of
the
first,
as
add
or
subtract
says)
6.5 ×
10
5
=
(3.6
+
or
−
2.9)
×
10
5
,
as
+
says
q
=
exponent
of
the
power
of
ten
at
or
below
S
(power
of
the
answer
=
exponent
of
the
power
of
ten
at
or
below
answer)
5
=
exponent
of
the
power
of
ten
at
or
below
6.5 ×
10
5
u
=
S
÷
10^
q
(front
of
the
answer
=
answer
÷
10power
of
the
answer)
6.5
=
6.5 ×
10
5
÷
10
5
Assumptions
•
Write both with the same power of ten first: the first number’s (the second’s front may pass 10).
•
Then add or subtract the fronts.
•
Rewrite the answer so its front is from 1 up to 10.
Step-by-step
WE KNOW
a = 3.6 (front of the first), n = 5 (power of the first), c = 2.9 (front of the second), k = 5 (power of the second), o: +
FIND
second front at the first power (g), answer (S), power of the answer (q), front of the answer (u)
Step 1 · Find second front at the first power (g)
Use g = c × 10ᵏ⁻ⁿ (second front at the first power = front of the second × 10ᵖᵒʷᵉʳᵒᶠᵗʰᵉˢᵉᶜᵒⁿᵈ⁻ᵖᵒʷᵉʳᵒᶠᵗʰᵉᶠⁱʳˢᵗ)
Rewrite the second front for the first number’s power of ten.
g
=
2.9
×
10^(5
−
5)
g
=
2.9 ×
10
0
g
=
2.9
×
1
g
=
2.9
Step 2 · Find answer (S)
Use S = (a + or − g) × 10ⁿ, as o says (answer = (front of the first + or − second front at the first power) × 10ᵖᵒʷᵉʳ of the first, as add or subtract says)
Add the fronts, keep the power of ten, then move the point so one digit is before it.
S
=
(3.6
+
2.9)
×
10
5
3.6 + 2.9 = 6.5
S
=
6.5 ×
10
5
Step 3 · Find power of the answer (q)
Use q = exponent of the power of ten at or below S (power of the answer = exponent of the power of ten at or below answer)
Count the places the point moves to leave one digit, not 0, before it.
q
=
exponent
of
the
power
of
ten
at
or
below
6.5 ×
10
5
q
=
5
Step 4 · Find front of the answer (u)
Use u = S ÷ 10q (front of the answer = answer ÷ 10ᵖᵒʷᵉʳ of the answer)
The front of the answer, from 1 up to 10.
u
=
6.5 ×
10
5
10
5
u
=
6.5 ×
10
5
100,000
u
=
6.5
Check
Put the numbers back in. Both sides should match.
2.9 = 2.9 × 105 − 5 ✓
(3.6
+
2.9)
×
10
5
=
6.5 ×
10
5
✓
5
=
exponent
of
the
power
of
ten
at
or
below
6.5 ×
10
5
✓
6.5
=
6.5 ×
10
5
÷
10
5
✓
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