Powers of Negative Numbers: Identify the greater value

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

• Step 1: Calculate $(-2)^7$
• Step 2: Calculate $-2^8$
• Step 3: Compare the two results

Now, let's work through each step:

Step 1: Calculate $(-2)^7$.

Using the power of negative numbers rule, $(-2)^7$ is a negative number because 7 is odd. We perform the calculation:

$(-2)^7 = - (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) = -128$.

Step 2: Calculate $-2^8$.

Here, $2^8$ is positive since 8 is an even number:

$2^8 = 256$.

But, note the negative sign in front: $-2^8 = -(2^8) = -256$.

Step 3: Compare $(-2)^7$ and $-2^8$:

We have $(-2)^7 = -128$ and $-2^8 = -256$. The comparison shows:

$-128 > -256$.

Therefore, the correct comparison is $(-2)^7 > -2^8$.

By following the steps and verifying calculations, we conclude that $$ $(-2)^7 > -2^8$.

First, let's evaluate $(-1)^{69}$. Since 69 is an odd number, $(-1)^{69} = -1$.

Next, evaluate $-(-1)^{79}$. The power 79 is also odd, so $(-1)^{79} = -1$. Applying the additional negative sign results in $-(-1) = 1$.

Now, we compare these two results. The expression $(-1)^{69} = -1$ and the expression $-(-1)^{79} = 1$.

Comparing the two: $-1$ is less than $1$.

Therefore, the relationship is $(-1)^{69} < -(-1)^{79}$.

The correct answer is $\lt$.

Let's address the problem by evaluating each expression separately:

Step 1: Evaluate $-2^4$.
Here, the expression represents the negative of $2^4$. The correct interpretation is $-(2^4)$.
Calculate $2^4 = 2 \times 2 \times 2 \times 2 = 16$.
Thus, $-2^4 = -16$.

Step 2: Evaluate $-(-2)^4$.
In this expression, $(-2)$ is raised to the power 4 first. Because 4 is an even number, $(-2)^4$ results in a positive value, specifically:
$(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16$.
Therefore, $-(-2)^4 = -16$.

Step 3: Compare the results.
We now compare the two outcomes:

• $-2^4 = -16$
• $-(-2)^4 = -16$

Both expressions evaluate to $-16$, hence they are equal.

Conclusion: $-2^4$ and $-(-2)^4$ are equal. Therefore, the relationship is $=$.

To compare $3^3$ and $-(-3)^3$, we will calculate each expression separately and then determine their relationship:

• Calculate $3^3$:

  $3^3$ means $3 \times 3 \times 3$.

  First, $3 \times 3 = 9$.

  Then, $9 \times 3 = 27$.

  Therefore, $3^3 = 27$.

• Calculate $-(-3)^3$:

  $(-3)^3$ means $(-3) \times (-3) \times (-3)$.

  First, $(-3) \times (-3) = 9$. (two negatives multiply to a positive)

  Then, $9 \times (-3) = -27$. (positive times negative is negative)

  Thus, $(-3)^3 = -27$.

  Considering the negative sign: $-(-3)^3 = -(-27) = 27$.

After calculating, we find that both expressions equal 27. Thus, they are equal.

The correct choice is:

$=$

To determine which is larger between $(-1)^{100}$ and $-1^{100}$, follow these steps:

• Step 1: Evaluate $(-1)^{100}$.
  Since 100 is an even number, $(-1)^{100}$ simplifies to $(-1)$ multiplied by itself 100 times. Even powers of -1 result in $1$, so $(-1)^{100} = 1$.
• Step 2: Evaluate $-1^{100}$.
  Notice that $-1^{100}$ is simply putting a negative sign in front of $1^{100}$. Since powers of 1 are always 1, $1^{100} = 1$, resulting in $-1^{100} = -1$.
• Step 3: Compare the results.
  From our calculations, $(-1)^{100} = 1$ and $-1^{100} = -1$. Comparing these, $1 > -1$.

Thus, the expression $(-1)^{100}$ is greater than $-1^{100}$.

Therefore, the correct comparison is $(-)^{100} > -1^{100}$.

The correct choice from the possible answers is $>$.

To solve this problem, follow these steps:

• Step 1: Calculate $-(2)^2$.
  Here, $(2)^2 = 4$, so $-(2)^2 = -4$.
• Step 2: Calculate $(-2)^3$.
  Since $(-2)^3 = (-2) \times (-2) \times (-2) = -8$.
• Step 3: Compare the results.
  We have $-(2)^2 = -4$ and $(-2)^3 = -8$. Comparing -4 and -8, we see that $-4 > -8$.

Since $-4$ is greater than $-8$, the symbol $>$ is correct.

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

To solve this problem, we'll compare the expressions $-(5^3)$ and $5^3$ by calculating each separately and then determining which is larger.

Step 1: Calculate $5^3$.
This is equal to $5 \times 5 \times 5 = 125$.

Step 2: Calculate $-(5^3)$.
Since $5^3 = 125$, applying the negative sign gives us $-(5^3) = -125$.

Step 3: Compare the values.
We have $-(5^3) = -125$ and $5^3 = 125$.
Clearly, $-125 < 125$.

Thus, the correct answer is that $-(5^3) \lt 5^3$.

The correct choice for this problem is $<$.

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

• Step 1: Evaluate the expression $-2 \cdot (3)^2$.
• Step 2: Evaluate the expression $-(-6)^2$.
• Step 3: Compare the results of Step 1 and Step 2.

Now, let's work through each step:

Step 1: Calculate $-2 \cdot (3)^2$.

First, find $(3)^2$. Since $3^2 = 9$, we have:

$-2 \cdot (3)^2 = -2 \cdot 9 = -18$.

Step 2: Calculate $-(-6)^2$.

First, find $(-6)^2$. Calculating the square gives:

$(-6)^2 = 36$.

Apply the negative sign: $-(-6)^2 = -36$.

Step 3: Compare the results.

We have $-2 \cdot (3)^2 = -18$ and $-(-6)^2 = -36$.

Since $-18$ is larger than $-36$, we conclude:

The expression $-2 \cdot (3)^2$ is greater than $-(-6)^2$.

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

To solve this problem, we must calculate both expressions step-by-step:

First, consider the expression $(( -4)^2)^2$:

• Evaluate $(-4)^2$:
  $(-4)^2 = (-4) \times (-4) = 16$.
• Now, evaluate $16^2$:
  $16^2 = 16 \times 16 = 256$.

The value of $(( -4)^2)^2$ is $256$.

Next, consider the expression $(-(4)^2)^2$:

• Evaluate $(4)^2$:
  $4^2 = 4 \times 4 = 16$.
• Now, consider the negative sign: $-(4)^2 = -(16) = -16$.
• Evaluate $(-16)^2$:
  $(-16)^2 = (-16) \times (-16) = 256$.

The value of $(-(4)^2)^2$ is $256$.

Therefore, the two expressions are equal.

Conclusion: The correct choice is $(=)$.

To solve this problem, we need to follow these steps:

• Step 1: Evaluate the first expression $(-(-3)^3)^2$
• Step 2: Evaluate the second expression $((2)^2)^4$
• Step 3: Compare the results from Steps 1 and 2

Now, let's proceed with these steps:

Step 1: Evaluate the expression $(-(-3)^3)^2$.
The inner expression is $(-3)^3$. Calculating this gives: $(-3)^3 = -27$ Next, we compute the expression $-(-3)^3$, which simplifies to: $-(-27) = 27$ Finally, we square this result: $(27)^2 = 729$ Thus, the value of the first expression is 729.

Step 2: Evaluate the expression $((2)^2)^4$.
First, calculate $(2)^2$: $(2)^2 = 4$ Next, raise this result to the fourth power: $(4)^4 = 256$ Thus, the value of the second expression is 256.

Step 3: Compare the two results from above:
We have $(-(-3)^3)^2 = 729$ and $((2)^2)^4 = 256$.

Since 729 is greater than 256, the expression $(-(-3)^3)^2$ is larger.

Thus, the correct answer is $\mathbf{>}$.