SME
Grade 11 Math · Problem type
Powers of the same base
Use this for “Solve 4⁶ = 8ˣ” by writing both sides as powers of 2.
Chart
Powers of 2: 2, 4, 8.
Amber bar: 2² = 4.
Exponent on the left: 6
Exponent on the right: 4
This is an example. Type your own values to start a new problem.
(
g
)
=
(
g
)
B₁
Left base
worked out (not typed)
B₂
Right base
worked out (not typed)
Formulas
B₁
=
g
p
4
=
2
2
B₂
=
g
q
8
=
2
3
q
×
x
=
p
×
m
3 × 4 = 2 × 6
Assumptions
•
Write both sides as powers of one base g: 4 = 2² and 8 = 2³.
•
A power of a power multiplies the exponents: (gᵖ)ᵐ = gᵖᵐ.
•
Equal powers of the same base have equal exponents, so p × m = q × x.
Step-by-step
WE KNOW
g = 2, p = 2, q = 3, m = 6
FIND
left base (B₁), right base (B₂), exponent on the right (x)
Step 1 · Find left base (B₁)
Use B₁ = gᵖ
The
left
base
as
a
power
of
g
.
B₁
=
2
2
B₁
=
4
Step 2 · Find right base (B₂)
Use B₂ = gq
The
right
base
as
a
power
of
g
.
B₂
=
2
3
B₂
=
8
Step 3 · Find exponent on the right (x)
Use q × x = p × m
Both
sides
are
powers
of
g
,
so
the
exponents
p
×
m
and
q
×
x
are
equal.
x
=
p
×
m
q
x
=
2
×
6
3
(
2
2
)
6
=
(
2
3
)ˣ
2
12
=
2
3
ˣ
3
x
=
12
x
=
4
Check
Put the numbers back in. Both sides should match.
4
=
2
2
✓
8
=
2
3
✓
3 × 4 = 2 × 6 ✓
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