Powers of Negative Numbers - Examples, Exercises and Solutions

To solve this problem, we need to evaluate expressions by applying the rules of exponents and the effects of parentheses on negative numbers:

• $(-3)^2$: When a negative number is squared, the result is positive. So, $(-3)^2$ means $-3 \times -3 = 9$.
• $-(-3)^2$: This means $-1 \times (-3 \times -3)$ because squaring a number negates the negative sign inside parentheses, resulting in $-9$.
• $-(3)^2$: This equals $-1 \times (3 \times 3) = -9$, as the negative sign is outside the squared value.
• $-3$: This is simply $-3$.

Only $(-3)^2$ equals 9, confirming it as the correct expression required by the problem.

Therefore, the solution to the problem is $(-3)^2$.

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 $-128$.

To determine which expression equals 36, we need to consider how squaring works with negative numbers:
Step 1: Consider the expression $(-6)^2$. This means that we take -6 and multiply it by itself:
$(-6) \times (-6) = 36$

Step 2: Consider the expression $-(6)^2$. Here, the square acts only on 6, not on the negative sign in front because of the absence of parentheses around -6:
$-(6 \times 6) = -36$

Therefore, the expression $(-6)^2$ correctly equals 36.

The correct choice that satisfies $36 =$ is $(-6)^2$.

To solve this problem, let's evaluate both given expressions to determine which results in 8.

• Step 1: Evaluate $(-2)^3$:
  $(-2)^3 = (-2) \times (-2) \times (-2)$.
  Multiplying across: $(-2) \times (-2) = 4$, and then $4 \times (-2) = -8$.
  Thus, $(-2)^3 = -8$.
• Step 2: Evaluate $-(-2)^3$:
  First, calculate $(-2)^3$ again: We already know $(-2)^3 = -8$.
  Now, apply the negative sign: $-(-8) = 8$.

Therefore, the expression that equals $8$ is $-(-2)^3$.

Thus, the correct expression that evaluates to 8 is $-(-2)^3$.

To solve this problem and express 64 as a power involving a negative number, we will follow these steps:

• Step 1: Recognize that we need to represent 64 using a base number squared. Since 64 is a perfect square, let's consider negative integers whose square equals 64.
• Step 2: The principal positive square root of 64 is 8. However, we are tasked with finding a negative number such that its square is 64.
• Step 3: If we have a negative integer, $(-8)$, and square it, we have: $(-8)^2 = (-8) \times (-8) = 64$.
• Step 4: Compare this with the expression $-(8)^2$, which results in $-64$ because the square applies only to 8, and the negative sign flips the result.

Therefore, the correct expression representing 64 with a negative base is $(-8)^2$, and among the answer choices provided, choice 1 is the correct one.

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 $-1$.

First let's recall the negative exponent rule:

$b^{-n}=\frac{1}{b^n}$We'll apply it to the expression we received:

$(-5)^{-3}=\frac{1}{(-5)^3}$Next let's recall the power rule for expressions in parentheses:

$(x\cdot y)^n=x^n\cdot y^n$And we'll apply it to the denominator of the expression we received:

$\frac{1}{(-5)^3}=\frac{1}{(-1\cdot5)^3}=\frac{1}{(-1)^3\cdot5^3}=\frac{1}{-1\cdot5^3}=-\frac{1}{5^3}=-\frac{1}{125}$In the first step, we expressed the negative number inside the parentheses in the denominator as a multiplication between a positive number and negative one, and then we used the power rule for expressions in parentheses to expand the parentheses, and then we simplified the expression.

Let's summarize the solution to the problem:

$(-5)^{-3}=\frac{1}{(-5)^3} =\frac{1}{-5^3}=-\frac{1}{125}$

Therefore, the correct answer is answer B.