To solve the equation
4
−
2
(
16
�
+
8
)
1
/
3
=
−
8
4−2(16x+8)
1/3
=−8, we follow these steps:
First, isolate the term involving the cube root:
4
−
2
(
16
�
+
8
)
1
/
3
=
−
8
4−2(16x+8)
1/3
=−8
Subtract 4 from both sides:
−
2
(
16
�
+
8
)
1
/
3
=
−
8
−
4
−2(16x+8)
1/3
=−8−4
−
2
(
16
�
+
8
)
1
/
3
=
−
12
−2(16x+8)
1/3
=−12
Divide both sides by -2 to isolate the cube root term:
(
16
�
+
8
)
1
/
3
=
−
12
/
−
2
(16x+8)
1/3
=−12/−2
(
16
�
+
8
)
1
/
3
=
6
(16x+8)
1/3
=6
Remove the cube root by cubing both sides:
(
16
�
+
8
)
=
6
3
(16x+8)=6
3
(
16
�
+
8
)
=
216
(16x+8)=216
Subtract 8 from both sides:
16
�
=
216
−
8
16x=216−8
16
�
=
208
16x=208
Finally, solve for
�
x by dividing by 16:
�
=
208
/
16
x=208/16
�
=
13
x=13
To check if
�
=
13
x=13 is an extraneous solution, we substitute it back into the original equation:
4
−
2
(
16
(
13
)
+
8
)
1
/
3
=
−
8
4−2(16(13)+8)
1/3
=−8
Simplifying:
4
−
2
(
208
+
8
)
1
/
3
=
−
8
4−2(208+8)
1/3
=−8
4
−
2
(
216
)
1
/
3
=
−
8
4−2(216)
1/3
=−8
4
−
2
(
6
)
=
−
8
4−2(6)=−8
4
−
12
=
−
8
4−12=−8
−
8
=
−
8
−8=−8
Since the equation is true, we conclude that
�
=
13
x=13 is not an extraneous solution.