Example 1 :
Evaluate :
(-1)^{4}
Solution :
Order of operations (PEMDAS) dictates that parentheses take precedence.
Here, the exponent 4 is an even number. So, the negative sign inside the parentheses will become positive.
When 1 is multiplied by itself any number of times, the result will be 1.
More clearly,
(-1)^{4 }= (-1) ⋅ (-1) (-1) ⋅ (-1)
(-1)^{4 }= 1
So, the value of (-1)^{4} is 1.
Example 2 :
Evaluate :
(-1)^{5}
Solution :
Order of operations (PEMDAS) dictates that parentheses take precedence.
Here, the exponent 5 is an odd number. So, the negative sign inside the parentheses will remain same.
When 1 is multiplied by itself any number of times, the result will be 1.
More clearly,
(-1)^{5 }= (-1) ⋅ (-1) ⋅ (-1) ⋅ (-1) ⋅ (-1)
(-1)^{4 }= -1
So, the value of (-1)^{5} is -1.
Example 3 :
Evaluate :
-(-3)^{3}
Solution :
Order of operations (PEMDAS) dictates that parentheses take precedence.
So, we have
-[(-3)^{3}] = -[(-3) ⋅ (-3) ⋅ (-3)]
-[(-3)^{3}] = -[-27]
-[(-3)^{3}] = 27
So, the value of (-3)^{3} is 27.
Example 4 :
Evaluate :
5^{0}
Solution :
Anything to the power zero is equal to 1.
So, we have
5^{0} = 1
So, the value of 5^{0 }is 1.
Example 5 :
Evaluate :
-3^{0}
Solution :
Anything to the power zero is equal to 1.
So, we have
-3^{0} = -1
So, the value of -3^{0 }is -1.
Example 6 :
Evaluate :
(-7)^{0}
Solution :
Order of operations (PEMDAS) dictates that parentheses take precedence.
Anything to the power zero is equal to 1.
So, we have
(-7)^{0} = 1
So, the value of (-7)^{0 }is -1.
Example 7 :
Evaluate :
5^{-3}
Solution :
Using laws of exponents, we have
5^{-3} = 1/5^{3}
5^{-3} = 1/125
So, the value of 5^{-3}^{ }is 1/125.
Example 8 :
Evaluate :
2^{3} ⋅ 3^{2 }⋅ (-1)^{5}
Solution :
In the above expression, first evaluate each term separately.
2^{3} = 2 ⋅ 2 ⋅ 2 = 8
3^{2} = 3 ⋅ 3 = 9
(-1)^{5 }= (-1) ⋅ (-1) ⋅ (-1) ⋅ (-1) ⋅ (-1) = -1
Now, we have
2^{3} ⋅ 3^{2 }⋅ (-1)^{5} = 8 ⋅ 9^{ }⋅ (-1)
2^{3} ⋅ 3^{2 }⋅ (-1)^{5} = - 72
So, the value 2^{3} ⋅ 3^{2 }⋅ (-1)^{5 }is -72.
Example 9 :
Evaluate :
(-1)^{4} ⋅ 3^{3 }⋅ 2^{2}
Solution :
In the above expression, first evaluate each term separately.
(-1)^{4 }= (-1) ⋅ (-1) ⋅ (-1) ⋅ (-1) = 1
3^{3} = 3 ⋅ 3 ⋅ 3 = 27
2^{2} = 2 ⋅ 2 = 4
Now, we have
(-1)^{4} ⋅ 3^{3 }⋅ 2^{2 }= 1 ⋅ 27^{ }⋅ 4
(-1)^{4} ⋅ 3^{3 }⋅ 2^{2 }= 108
So, the value (-1)^{4} ⋅ 3^{3 }⋅ 2^{2}^{ }is 108.
Example 10 :
If x^{2}y^{3} = 10 and x^{3}y^{2} = 8, then find the value of x^{5}y^{5}.
Solution :
x^{2}y^{3} = 10 ----(1)
x^{3}y^{2} = 8 ----(2)
Multiply (1) and (2) :
(1) ⋅ (2) ----> (x^{2}y^{3}) ⋅ (x^{3}y^{2}) = 10 ⋅ 8
x^{5}y^{5} = 80
So, the value x^{5}y^{5 }is 80.
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