SME
Grade 8 Math · Problem type
A power of a power
Use this for “Write (10⁷)² with a single exponent.”
Diagram
(2³)⁴ = 2¹² = 4,096
4 groups of 3 factors: 4 × 3 = 12 factors of 2.
This is an example. Type your own values to start a new problem.
(
b
)
=
b
=
P
Formulas
k
=
m
×
n
(exponent
of
the
answer
=
inside
exponent
×
outside
exponent)
12 = 3 × 4
P
=
b
^
k
(value
of
the
answer
=
baseexponent
of
the
answer)
4,096
=
2
12
Assumptions
•
(2³)⁴ is 2³ used as a factor 4 times.
•
Each copy has 3 factors of 2, so there are 3 × 4 of them: multiply the exponents.
Step-by-step
WE KNOW
b = 2 (base), m = 3 (inside exponent), n = 4 (outside exponent)
FIND
exponent of the answer (k), value of the answer (P)
Step 1 · Find exponent of the answer (k)
Use k = m × n (exponent of the answer = inside exponent × outside exponent)
A power of a power multiplies the exponents.
k
=
3
×
4
k
=
12
Step 2 · Find value of the answer (P)
Use P = bᵏ (value of the answer = baseᵉˣᵖᵒⁿᵉⁿᵗ of the answer)
Multiply the base by itself as many times as the exponent says.
P
=
2
12
P
=
4,096
Check
Put the numbers back in. Both sides should match.
12 = 3 × 4 ✓
2
12
=
4,096
✓
Topic check
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