SME
Grade 9 Math · Problem type
Solve by writing the same base
Use this for “Solve 4⁶ = 8ˣ.”
Table
Solution (x)
q to the power x, for the table (Q)
This is an example. Type your own values to start a new problem.
p
=
q
(
b
)
=
(
b
)
The second line writes both bases as powers of b. Equal bases need equal exponents.
e
Exponent of b on each side
worked out (not typed)
R
Each side’s value, b to the power e
worked out (not typed)
Formulas
b
=
the
common
base
of
p
and
q
2 = the common base of 4 and 8
s
=
the
power
of
b
that
makes
p
2 = the power of 2 that makes 4
t
=
the
power
of
b
that
makes
q
3 = the power of 2 that makes 8
e
=
s
×
a
12 = 2 × 6
x
=
e
t
4
=
12
3
R
=
b
e
4,096
=
2
12
Assumptions
•
Write both sides as powers of one base: 4 = 2² and 8 = 2³.
•
A power of a power multiplies the exponents: (2²)⁶ = 2¹².
•
When the bases are equal, the exponents are equal: solve 3x = 12.
Step-by-step
WE KNOW
p = 4, a = 6, q = 8
FIND
common base (b), p as a power of b (s), q as a power of b (t), exponent of b on each side (e), solution (x), each side’s value, b to the power e (R)
Step 1 · Find common base (b)
Use b = the common base of p and q
Find the smallest number both bases are powers of.
b
=
the
common
base
of
4
and
8
b
=
2
Step 2 · Find p as a power of b (s)
Use s = the power of b that makes p
Write
the
left
base
as
a
power
of
b
.
s
=
the
power
of
2
that
makes
4
4
=
2
2
s
=
2
Step 3 · Find q as a power of b (t)
Use t = the power of b that makes q
Write
the
right
base
as
a
power
of
b
too.
t
=
the
power
of
2
that
makes
8
8
=
2
3
t
=
3
Step 4 · Find exponent of b on each side (e)
Use e = s × a
A
power
of
a
power
multiplies
the
exponents:
the
left
side
is
b
to
the
power
sa.
e
=
2
×
6
(
2
2
)
6
=
2
12
e
=
12
Step 5 · Find solution (x)
Use x = e ÷ t
The
right
side
is
b
to
the
power
tx.
The
bases
match,
so
tx
=
e
:
divide
by
t
.
x
=
12
3
3
x
=
12
x
=
4
Step 6 · Find each side’s value, b to the power e (R)
Use R = bᵉ
Check: both sides are this number.
R
=
2
12
R
=
4,096
Check
Put the numbers back in. Both sides should match.
2 = the common base of 4 and 8 ✓
2 = the power of 2 that makes 4 ✓
3 = the power of 2 that makes 8 ✓
12 = 2 × 6 ✓
4
=
12
3
✓
4
6
=
4,096
✓
Topic check
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