Powers of Negative Numbers: Combination with exercise

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

The first step is to evaluate each component:

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

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

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

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

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

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

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

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

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

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

Therefore, the solution to the problem is $0$.

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

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

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

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

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

Now, let's work through each step:

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

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

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

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

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

Step 2: Compute $2^3$.

$2^3 = 2 \times 2 \times 2$

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

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

Therefore, $2^3 = 8$.

Step 3: Add the two results together.

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

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

Therefore, the solution to the problem is $0$.

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

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

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

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

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: $(-2)^2$. This means we first square $-2$, which results in $4$ because $(-2) \times (-2) = 4$.
• Step 2: Now, consider the negative sign outside the squared term. So, evaluate $-(-2)^2$, which simplifies to $-4$.
• Step 3: Next, we need to square $-4$: $(-4)^2 = (-4) \times (-4) = 16$.
• Step 4: Now, calculate the second part of the expression, $2^3$, which equals $8$ because $2 \times 2 \times 2 = 8$.
• Step 5: Finally, subtract the result from Step 4 from the result of Step 3: $16 - 8 = 8$.

Thus, the solution to the problem is $8$.

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

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

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

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

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

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