Powers of Negative Numbers: Applying the formula

When we have a negative number raised to a power, the location of the minus sign is very important.

If the minus sign is inside or outside the parentheses, the result of the exercise can be completely different.

When the minus sign is inside the parentheses, our exercise will look like this:

(-8)*(-8)=

Since we know that minus times minus is actually plus, the result will be positive:

(-8)*(-8)=64

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

• Step 1: Calculate $(2)^2$
• Step 2: Apply the negative sign

Now, let's work through each step:
Step 1: Calculate $(2)^2$. This is equal to $2 \times 2 = 4$.
Step 2: Apply the negative sign: The expression $-(2)^2$ now becomes $-4$.

Therefore, the value of the expression $-(2)^2$ is $-4$.

This matches choice 4, which is $-4$.

To solve for $(-2)^7$, follow these steps:

• Step 1: Identify the base and the exponent given in the expression, which are $-2$ and $7$, respectively.
• Step 2: Recognize that since the exponent is $7$, which is an odd number, the result of the power will remain negative: $(-2)^7$ will be $- (2^7)$.
• Step 3: Compute $2^7$. This involves multiplying $2$ by itself $7$ times:
  $2 \times 2 = 4$
  $4 \times 2 = 8$
  $8 \times 2 = 16$
  $16 \times 2 = 32$
  $32 \times 2 = 64$
  $64 \times 2 = 128$
  Thus, $2^7 = 128$.
• Step 4: Apply the negative sign to the result of $2^7$, resulting in $-128$.

Therefore, the value of $(-2)^7$ is $-128$.

The given problem asks us to evaluate the expression $-(7)^2$. To solve this, we must correctly handle the operations of exponentiation and negation.

Firstly, examine $(7)^2$:
- $(7)^2$ means multiplying 7 by itself.
- Calculating this gives: $7 \times 7 = 49$.

Next, apply the negative sign to the result:
- The expression $-(7)^2$ indicates that we apply the negative sign to the result of $(7)^2$.
- Therefore, multiply the result by $-1$:
$-1 \times 49 = -49$.

Thus, the correct evaluation of the expression is $-49$.

Thus, the solution to this problem is $-49$.

To solve the problem of evaluating $-(-6)^2$, we will follow these steps:

• Step 1: Calculate the square of $-6$.
• Step 2: Apply the negative sign to the result obtained from the first step.

Let's work through these steps:
Step 1: Calculate $(-6)^2$. We know that when squaring a negative number, the result becomes positive: $(-6) \times (-6) = 36$.
Step 2: Now apply the negative sign to this result. The expression is $-(36)$, which equals $-36$.

Therefore, the solution to the problem is $-36$.

To solve the expression $-(-2)^3$, we need to first calculate the inner power and then apply the outer negative sign.

• Step 1: Calculate $(-2)^3$.
  Since $(-2)$ is raised to the power of 3, we perform the multiplication: $(-2) \times (-2) \times (-2)$.
  $(-2) \times (-2) = 4$.
  Continuing, $4 \times (-2) = -8$.
  Thus, $(-2)^3 = -8$.
• Step 2: Apply the outer negative sign.
  We have $-(-2)^3 = -(-8)$.
  According to arithmetic rules, a negative times a negative becomes positive, so $-(-8) = 8$.

Therefore, the solution to the problem is $8$, which matches choice number 2.

To solve the problem, we need to evaluate the expression $-(-1)^{100}$.

• Step 1: Assess the exponent in $(-1)^{100}$. Since $100$ is an even number, the property of powers of $-1$ tells us that $(-1)^{100} = 1$.
• Step 2: Now, consider the entire expression. We have an outer negation: $-(+1)$. Arithmetic operation of negation results in $-1$.

Therefore, the solution to the problem is $-1$.

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

• Step 1: Determine the nature of the exponent (odd or even).
• Step 2: Apply the specific rule for the power of $-1$ based on the exponent's nature.

Now, let's work through each step:
Step 1: The exponent is $99$, which is odd.
Step 2: For the power of $-1$, the rule states that if the exponent is odd, $(-1)^n = -1$. Thus, $(-1)^{99} = -1$.

Therefore, the solution to the problem is $-1$.

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

• Step 1: Recognize the order of operations.
• Step 2: Evaluate the exponent first before applying any operations outside the exponent.
• Step 3: Apply the negative sign to the result of the squared value.

Now, let's work through each step:
Step 1: The expression $-6^2$ involves squaring the number 6. According to the order of operations, we compute exponents before multiplying by -1.
Step 2: This means we first calculate $6^2$, which is equal to 36.
Step 3: After evaluating the square, apply the negative sign: $-6^2 = -(6^2) = -36$.

Therefore, the solution to the problem is $-36$.

To solve this problem, we'll evaluate the expression $-(-1)^{80}$.

• Step 1: Evaluate $(-1)^{80}$.

Since the exponent 80 is an even number, by applying the rule for negative powers, $(-1)^{80} = 1$.

• Step 2: Apply the negation.

The expression is $-(-1)^{80}$, which simplifies to $-1$, because negating 1 results in $-1$.

Therefore, the solution to the problem is $-1$.

To solve the problem $(-3)^4$, we'll perform the following computations step-by-step:

• Step 1: Consider $(-3) \times (-3)$. This equals $9$ because multiplying two negative numbers results in a positive number.
• Step 2: Use the result from Step 1 and multiply it by $(-3)$: $9 \times (-3) = -27$.
• Step 3: Finally, multiply the result from Step 2 by $(-3)$: $-27 \times (-3) = 81$.

The properties of exponents ensure that $(-3)^4$ results in a positive number because the exponent is even.

Therefore, the solution to the problem is $\textbf{81}$. The correct answer corresponds to choice 3.

To solve this problem, we'll methodically approach it by following these steps:

• Step 1: Simplify the expression inside the parentheses.
• Step 2: Evaluate the power applied to the result from Step 1.

Now, let's execute each step in detail:

Step 1: Simplify the expression inside the parentheses $(-(2)^2)$.

Calculate $(2)^2$:

$2^2 = 4$

Substitute back into the expression:

$-(2)^2 = -4$

Step 2: Evaluate the expression $(-4)^2$.

Calculate the square:

$(-4)^2 = 16$

The solution to the problem is $16$, which corresponds to choice 2.

To solve the expression $-(-(3)^2)^2$, we will strictly follow the order of operations, PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Step 1: Evaluate the inner exponent.
  We compute $(3)^2 = 9$.
• Step 2: Apply the first negative sign.
  The expression becomes $-((3)^2) = -(9) = -9$.
• Step 3: Apply the second negative sign and exponent.
  Now, simplify $-(-9)^2$.
  First, evaluate the exponent: $(-9)^2 = (-9) \times (-9) = 81$.
  Then, apply the negative sign: $-81$.

Therefore, the solution to the problem is $-81$.

To solve the given problem, we'll use rules of exponents and follow these steps:

• Step 1: Calculate the inner power. Simplify $(-2)^2$.
• Step 2: Calculate the result by taking the square of the previous result.

Here's the detailed breakdown:

Step 1: Determine $(-2)^2$.
Using the property of exponents, $(-2)^2 = (-2) \times (-2) = 4$.

Step 2: Now, square the result from Step 1.
You are to calculate $4^2$.
$4^2 = 4 \times 4 = 16$.

Therefore, the solution to the problem is $\boxed{16}$.

To solve $-(-5)^3$, we proceed as follows:

• Step 1: Calculate $(-5)^3$. Since the exponent is 3, which is odd, the result will be negative. Specifically:
  $(-5)^3 = (-5) \times (-5) \times (-5)$.
• Step 2: Compute the multiplication:
  $(-5) \times (-5) = 25$ (as multiplying two negatives gives a positive).
  $25 \times (-5) = -125$ (as multiplying positive by negative gives a negative).
• Step 3: Now apply the outer negative to this result:
  $-(-125) = 125$.

Therefore, the solution to the problem is $125$.