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Let’s check out your problem:
If
2
a
=
2
5
7
2^a=\sqrt[7]{2^5}
2
a
=
7
2
5
, what is the value of
a
a
a
?
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Math Problems
Grade 8
Solve equations with variable exponents
Full solution
Q.
If
2
a
=
2
5
7
2^a=\sqrt[7]{2^5}
2
a
=
7
2
5
, what is the value of
a
a
a
?
Identify Given Equation:
Identify the given equation and the goal.
\newline
We are given that
2
a
=
2
5
7
2^a = \sqrt[7]{2^5}
2
a
=
7
2
5
, and we need to find the value of
a
a
a
.
Express Seventh Root:
Express the seventh root as an exponent.
\newline
The seventh root of
2
5
2^5
2
5
can be written as
(
2
5
)
1
7
(2^5)^{\frac{1}{7}}
(
2
5
)
7
1
.
Apply Power Rule:
Apply the power rule for exponents, which states that
(
a
m
)
n
=
a
m
∗
n
(a^m)^n = a^{m*n}
(
a
m
)
n
=
a
m
∗
n
.
\newline
So,
(
2
5
)
1
/
7
(2^5)^{1/7}
(
2
5
)
1/7
becomes
2
5
∗
(
1
/
7
)
2^{5*(1/7)}
2
5
∗
(
1/7
)
.
Multiply Exponents:
Multiply the exponents to simplify the expression.
\newline
5
×
(
1
7
)
=
5
7
5 \times \left(\frac{1}{7}\right) = \frac{5}{7}
5
×
(
7
1
)
=
7
5
.
\newline
So,
2
5
×
(
1
7
)
2^{5\times\left(\frac{1}{7}\right)}
2
5
×
(
7
1
)
simplifies to
2
5
7
2^{\frac{5}{7}}
2
7
5
.
Set Equal to
2
a
2^a
2
a
:
Set the simplified expression equal to
2
a
2^a
2
a
. We have
2
a
=
2
5
/
7
2^a = 2^{5/7}
2
a
=
2
5/7
.
Determine Value of
a
a
a
:
Since the bases are the same, the exponents must be equal.
\newline
Therefore,
a
=
5
7
a = \frac{5}{7}
a
=
7
5
.
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\newline
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/
(
2
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(B)
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)
/
(
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9
)
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(
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)
/
(
8
9
)
\newline
(C)
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1
)
/
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(
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b
7
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(
4
5
b
7
)
−
3
=
?
\newline
Choose
1
1
1
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\newline
(A)
b
21
4
15
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4
15
b
21
\newline
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b
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1
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\newline
(A)
1
3
9
⋅
6
18
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3
9
⋅
6
18
1
\newline
(B)
6
18
3
9
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3
9
6
18
\newline
(C)
3
9
6
18
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6
18
3
9
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