ONE EIGHTH CONTEST 2025
Ghana National College - 65 points
Tema Secondary School - 33 points
Effiduase SHCS - 28 points
ONE EIGHTH CONTEST 2025
Mankranso SHS - 43 points
St Paul's SHS, Denu - 23 points
Adonten SHS - 14 points
ONE EIGHTH CONTEST 2025
St. Thomas Aquinas SHS - 51 points
Tamale SHS - 37 points
O'Reilly SHS - 25 points
ROUND 1
PREAMBLE
Solve for $x$ from the given equation
1. $(x-2)^{x-1}=1$
ANSWER: $x=1$ or $x=3$
SOLUTION
An expression $a^b=1$ when the base is $1$ or the exponent is $0$.
Case 1: the base equals $1$.
$x-2=1$
$x=1+2$
$x=3$
Case 2: the exponent equals $0$.
$(x-2)^{x-1}=(x-2)^0$
$x-1=0$
$x=1$
$x=1$ or $x=3$
2. $(x+2)^{x-2}=1$
ANSWER: $x=-1$ or $x=2$
SOLUTION
An expression $a^b=1$ when the base is $1$ or the exponent is $0$.
Case 1: the base equals $1$.
$x+2=1$
$x=1-2$
$x=-1$
Case 2: the exponent equals $0$.
$(x+2)^{x-2}=(x+2)^0$
$x-2=0$
$x=2$
$x=-1$ or $x=2$
3. $(2x-1)^{x-3}=1$
ANSWER: $x=1$ or $x=3$
SOLUTION
An expression $a^b=1$ when the base is $1$ or the exponent is $0$.
Case 1: the base equals $1$.
$2x-1=1$
$2x=1+1$
$2x=2$
$x=\dfrac{2}{2}$
$x=1$
Case 2: the exponent equals $0$.
$(2x-1)^{x-3}=(2x-1)^0$
$x-3=0$
$x=3$
$x=1$ or $x=3$
PRACTICE QUESTIONS
Solve for $x$ from the given equation.
1. $(x-4)^{x+2}=1$
ANSWER: $x=-2$, $x=5$
SOLUTION
An expression $a^b=1$ when the base is $1$, or the exponent is $0$, or the base is $-1$ with an even exponent.
Case 1: the base equals $1$.
$x-4=1$
$x=5$
Case 2: the exponent equals $0$.
$x+2=0$
$x=-2$
Case 3: the base equals $-1$.
$x-4=-1$
$x=3$
The exponent is $5$, which is odd, so $x=3$ is rejected.
$x=-2,\ 5$
2. $(x+1)^{x-3}=1$
ANSWER: $x=0$, $x=3$
SOLUTION
An expression $a^b=1$ when the base is $1$, or the exponent is $0$, or the base is $-1$ with an even exponent.
Case 1: the base equals $1$.
$x+1=1$
$x=0$
Case 2: the exponent equals $0$.
$x-3=0$
$x=3$
Case 3: the base equals $-1$.
$x+1=-1$
$x=-2$
The exponent is $-5$, which is odd, so $x=-2$ is rejected.
$x=0,\ 3$
3. $(2x-3)^{x-5}=1$
ANSWER: $x=1$, $x=2$, $x=5$
SOLUTION
An expression $a^b=1$ when the base is $1$, or the exponent is $0$, or the base is $-1$ with an even exponent.
Case 1: the base equals $1$.
$2x-3=1$
$x=2$
Case 2: the exponent equals $0$.
$x-5=0$
$x=5$
Case 3: the base equals $-1$.
$2x-3=-1$
$x=1$
The exponent is $-4$, which is even, so $x=1$ is valid.
$x=1,\ 2,\ 5$
4. $(x-5)^{x+1}=1$
ANSWER: $x=-1$, $x=6$
SOLUTION
An expression $a^b=1$ when the base is $1$, or the exponent is $0$, or the base is $-1$ with an even exponent.
Case 1: the base equals $1$.
$x-5=1$
$x=6$
Case 2: the exponent equals $0$.
$x+1=0$
$x=-1$
Case 3: the base equals $-1$.
$x-5=-1$
$x=4$
The exponent is $5$, which is odd, so $x=4$ is rejected.
$x=-1,\ 6$
5. $(2x+1)^{x-4}=1$
ANSWER: $x=0$, $x=4$
SOLUTION
An expression $a^b=1$ when the base is $1$, or the exponent is $0$, or the base is $-1$ with an even exponent.
Case 1: the base equals $1$.
$2x+1=1$
$x=0$
Case 2: the exponent equals $0$.
$x-4=0$
$x=4$
Case 3: the base equals $-1$.
$2x+1=-1$
$x=-1$
The exponent is $-5$, which is odd, so $x=-1$ is rejected.
$x=0,\ 4$
6. $(x+3)^{x-1}=1$
ANSWER: $x=-2$, $x=1$
SOLUTION
An expression $a^b=1$ when the base is $1$, or the exponent is $0$, or the base is $-1$ with an even exponent.
Case 1: the base equals $1$.
$x+3=1$
$x=-2$
Case 2: the exponent equals $0$.
$x-1=0$
$x=1$
Case 3: the base equals $-1$.
$x+3=-1$
$x=-4$
The exponent is $-5$, which is odd, so $x=-4$ is rejected.
$x=-2,\ 1$