Examples with solutions for Powers of Negative Numbers: Combination with exercise

Exercise #1

(1)99(1)9= (-1)^{99}\cdot(-1)^9=

Video Solution

Step-by-Step Solution

To solve this problem, we need to evaluate the expression (1)99(1)9 (-1)^{99} \cdot (-1)^9 .

The first step is to evaluate each component:

  • Step 1: Evaluate (1)99 (-1)^{99} .
    Since 99 is an odd number, (1)99=1(-1)^{99} = -1. This is because any odd power of 1-1 results in 1-1.
  • Step 2: Evaluate (1)9 (-1)^9 .
    Similarly, since 9 is also an odd number, (1)9=1(-1)^9 = -1. Again, an odd exponent means the result is 1-1.

Step 3: Multiply the results from step 1 and step 2:
(1)99(1)9=(1)(1)=1 (-1)^{99} \cdot (-1)^9 = (-1) \cdot (-1) = 1 .

Thus, the value of the expression is 1\boxed{1}.

Answer

1 1

Exercise #2

33+(3)3= -3^3+(-3)^3=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Calculate 33 -3^3 and (3)3 (-3)^3 .
  • Step 2: Add the results from both calculations.

Now, let's break this down further:
Step 1: First, compute 33 -3^3 .
The expression 33 -3^3 should be interpreted as (33)- (3^3). This means we first calculate 33 3^3 , which is 3×3×3=27 3 \times 3 \times 3 = 27 . The negative sign in front gives us 27-27.
Next, calculate (3)3 (-3)^3 .
Here, the base is 3-3, so we calculate (3)×(3)×(3)(-3) \times (-3) \times (-3). This gives us:
- Multiply the first two factors: (3)×(3)=9(-3) \times (-3) = 9.
- Multiply the result by the last factor: 9×(3)=279 \times (-3) = -27.
So, (3)3=27(-3)^3 = -27.

Step 2: Add the results 27+(27)-27 + (-27).
This computation is 2727=54-27 - 27 = -54.

Therefore, the solution to the problem is 54-54.

Answer

54 -54

Exercise #3

62(6)2= 6^2-(-6)^2=

Video Solution

Step-by-Step Solution

To solve the expression 62(6)2 6^2 - (-6)^2 , let's follow these steps:

  • Step 1: Calculate 62 6^2 .
    62=6×6=36 6^2 = 6 \times 6 = 36 .
  • Step 2: Calculate (6)2(-6)^2.
    (6)2=(6)×(6)=36(-6)^2 = (-6) \times (-6) = 36 . Note that squaring a negative number results in a positive value.
  • Step 3: Subtract the results of Step 1 and Step 2.
    3636=0 36 - 36 = 0 .

Therefore, the solution to the problem is 0 0 .

Answer

0 0

Exercise #4

(1)10022= -(-1)^{100}\cdot2^2=

Video Solution

Step-by-Step Solution

To solve this problem, we need to evaluate (1)10022-(-1)^{100}\cdot2^2.

  • Step 1: Calculate (1)100(-1)^{100}. Since 100 is an even number, (1)100=1(-1)^{100} = 1.
  • Step 2: Calculate 222^2. This gives us 44.
  • Step 3: Multiply the results from the first two steps: 14=41 \cdot 4 = 4.
  • Step 4: Apply the negative sign: 4-4.

Thus, when evaluating the expression, we find that the correct result is 4-4.

Answer

4 -4

Exercise #5

(2)3+23= (-2)^3+2^3=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Compute (2)3(-2)^3.
  • Step 2: Compute 232^3.
  • Step 3: Add the two results.

Now, let's work through each step:

Step 1: (2)3(-2)^3 means multiplying 2-2 by itself three times.

(2)3=(2)×(2)×(2)(-2)^3 = (-2) \times (-2) \times (-2)

Start by multiplying the first two 2-2's: (2)×(2)=4(-2) \times (-2) = 4 (since the product of two negatives is positive).

Next, multiply the result by the third 2-2: 4×(2)=84 \times (-2) = -8.

Therefore, (2)3=8(-2)^3 = -8.

Step 2: Compute 232^3.

23=2×2×22^3 = 2 \times 2 \times 2

Multiply the first two 2's: 2×2=42 \times 2 = 4.

Then multiply the result by the third 2: 4×2=84 \times 2 = 8.

Therefore, 23=82^3 = 8.

Step 3: Add the two results together.

We have (2)3+23=8+8(-2)^3 + 2^3 = -8 + 8.

Calculate the sum: 8+8=0-8 + 8 = 0.

Therefore, the solution to the problem is 00.

Answer

0 0

Exercise #6

(5)3+52= (-5)^3+5^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll calculate each part of the expression separately:

  • Step 1: Calculate (5)3 (-5)^3 . Since 3 3 is odd, (5)3=(5)(5)(5)=125 (-5)^3 = (-5) \cdot (-5) \cdot (-5) = -125 .
  • Step 2: Calculate 52 5^2 . As 2 2 is an even number, 52=55=25 5^2 = 5 \cdot 5 = 25 .
  • Step 3: Add the results of the two calculations: 125+25 -125 + 25 .

Now, let's work through each step:
Step 1: We calculated that (5)3=125 (-5)^3 = -125 .
Step 2: We found that 52=25 5^2 = 25 .
Step 3: Adding these results: 125+25=100 -125 + 25 = -100 .

Therefore, the solution to the problem is 100 -100 .

Answer

100 -100

Exercise #7

((2)2)223= (-(-2)^2)^2-2^3=

Video Solution

Step-by-Step Solution

To solve this problem, we need to carefully apply the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Let's evaluate the expression step-by-step:

  • Step 1: Begin with the expression inside the innermost parentheses: (2)2(-2)^2. This means we first square 2-2, which results in 44 because (2)×(2)=4(-2) \times (-2) = 4.
  • Step 2: Now, consider the negative sign outside the squared term. So, evaluate (2)2-(-2)^2, which simplifies to 4-4.
  • Step 3: Next, we need to square 4-4: (4)2=(4)×(4)=16(-4)^2 = (-4) \times (-4) = 16.
  • Step 4: Now, calculate the second part of the expression, 232^3, which equals 88 because 2×2×2=82 \times 2 \times 2 = 8.
  • Step 5: Finally, subtract the result from Step 4 from the result of Step 3: 168=816 - 8 = 8.

Thus, the solution to the problem is 8 8 .

Answer

8 8

Exercise #8

(2)4+(2)2= -(-2)^4+(-2)^2=

Video Solution

Step-by-Step Solution

To solve this problem, let's evaluate the expression (2)4+(2)2-(-2)^4 + (-2)^2 step-by-step:

  • Step 1: Calculate (2)4(-2)^4.
    Since the exponent is even, (2)4=((2)×(2)×(2)×(2))(-2)^4 = ((-2) \times (-2) \times (-2) \times (-2)).
    Calculating more explicitly:
    (2)×(2)=4(-2) \times (-2) = 4,
    4×(2)=84 \times (-2) = -8,
    8×(2)=16-8 \times (-2) = 16.
    Thus, (2)4=16(-2)^4 = 16.

  • Step 2: Negate the result from step 1.
    (2)4=(16)=16-(-2)^4 = -(16) = -16.

  • Step 3: Calculate (2)2(-2)^2.
    Since the exponent is even, (2)2=(2)×(2)=4(-2)^2 = (-2) \times (-2) = 4.

  • Step 4: Add the results from step 2 and step 3.
    16+4=12-16 + 4 = -12.

Therefore, the final result is 12 \mathbf{-12} .

Answer

12 -12