Examples with solutions for Powers of Negative Numbers: Identify the greater value

Exercise #1

Which is larger?

(2)728 (-2)^7⬜-2^8

Video Solution

Step-by-Step Solution

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

  • Step 1: Calculate (2)7 (-2)^7
  • Step 2: Calculate 28 -2^8
  • Step 3: Compare the two results

Now, let's work through each step:

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

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

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

Step 2: Calculate 28 -2^8 .

Here, 28 2^8 is positive since 8 is an even number:

28=2562^8 = 256.

But, note the negative sign in front: 28=(28)=256-2^8 = -(2^8) = -256.

Step 3: Compare (2)7(-2)^7 and 28-2^8:

We have (2)7=128(-2)^7 = -128 and 28=256-2^8 = -256. The comparison shows:

128>256-128 > -256.

Therefore, the correct comparison is (2)7>28(-2)^7 > -2^8.

By following the steps and verifying calculations, we conclude that (2)7>28 (-2)^7 > -2^8 .

Answer

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Exercise #2

Which is larger?

(53)53 -(5^3)⬜5^3

Video Solution

Step-by-Step Solution

To solve this problem, we'll compare the expressions (53) -(5^3) and 53 5^3 by calculating each separately and then determining which is larger.

Step 1: Calculate 53 5^3 .
This is equal to 5×5×5=125 5 \times 5 \times 5 = 125 .

Step 2: Calculate (53) -(5^3) .
Since 53=125 5^3 = 125 , applying the negative sign gives us (53)=125 -(5^3) = -125 .

Step 3: Compare the values.
We have (53)=125 -(5^3) = -125 and 53=125 5^3 = 125 .
Clearly, 125<125-125 < 125.

Thus, the correct answer is that (53)<53 -(5^3) \lt 5^3 .

The correct choice for this problem is < .

Answer

<

Exercise #3

Which is larger?

(2)2(2)3 -(2)^2⬜(-2)^3

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

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

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

Therefore, the solution to the problem is > > .

Answer

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Exercise #4

Which is larger?

(1)69(1)79 (-1)^{69}⬜-(-1)^{79}

Video Solution

Step-by-Step Solution

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

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

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

Comparing the two: 1-1 is less than 11.

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

The correct answer is <\lt.

Answer

<

Exercise #5

Which is larger?

24(2)4 -2^4⬜-(-2)^4

Video Solution

Step-by-Step Solution

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

Step 1: Evaluate 24 -2^4 .
Here, the expression represents the negative of 242^4. The correct interpretation is (24) -(2^4) .
Calculate 24=2×2×2×2=16 2^4 = 2 \times 2 \times 2 \times 2 = 16 .
Thus, 24=16 -2^4 = -16 .

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

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

  • 24=16 -2^4 = -16
  • (2)4=16-(-2)^4 = -16

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

Conclusion: 24 -2^4 and (2)4-(-2)^4 are equal. Therefore, the relationship is = = .

Answer

= =

Exercise #6

Which is larger?

2(3)2(6)2 -2\cdot(3)^2⬜-(-6)^2

Video Solution

Step-by-Step Solution

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

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

Now, let's work through each step:

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

First, find (3)2 (3)^2 . Since 32=9 3^2 = 9 , we have:

2(3)2=29=18-2 \cdot (3)^2 = -2 \cdot 9 = -18.

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

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

(6)2=36(-6)^2 = 36.

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

Step 3: Compare the results.

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

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

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

Therefore, the solution to the problem is > .

Answer

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Exercise #7

Which is larger?

(1)1001100 (-1)^{100}⬜-1^{100}

Video Solution

Step-by-Step Solution

To determine which is larger between (1)100 (-1)^{100} and 1100-1^{100} , follow these steps:

  • Step 1: Evaluate (1)100 (-1)^{100} .
    Since 100 is an even number, (1)100 (-1)^{100} simplifies to (1)(-1) multiplied by itself 100 times. Even powers of -1 result in 11, so (1)100=1 (-1)^{100} = 1 .
  • Step 2: Evaluate 1100-1^{100} .
    Notice that 1100-1^{100} is simply putting a negative sign in front of 11001^{100}. Since powers of 1 are always 1, 1100=1 1^{100} = 1 , resulting in 1100=1-1^{100} = -1 .
  • Step 3: Compare the results.
    From our calculations, (1)100=1 (-1)^{100} = 1 and 1100=1-1^{100} = -1 . Comparing these, 1>11 > -1.

Thus, the expression (1)100 (-1)^{100} is greater than 1100-1^{100} .

Therefore, the correct comparison is ()100>1100(-)^{100} > -1^{100}.

The correct choice from the possible answers is > > .

Answer

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Exercise #8

Which is larger?

33(3)3 3^3⬜-(-3)^3

Video Solution

Step-by-Step Solution

To compare 333^3 and (3)3-(-3)^3, we will calculate each expression separately and then determine their relationship:

  • Calculate 333^3:

    333^3 means 3×3×33 \times 3 \times 3.

    First, 3×3=93 \times 3 = 9.

    Then, 9×3=279 \times 3 = 27.

    Therefore, 33=273^3 = 27.

  • Calculate (3)3-(-3)^3:

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

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

    Then, 9×(3)=279 \times (-3) = -27. (positive times negative is negative)

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

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

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

The correct choice is:

= =

Answer

= =

Exercise #9

Which is larger?

((4)2)2((4)2)2 ((-4)^2)^2⬜(-(4)^2)^2

Video Solution

Step-by-Step Solution

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

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

  • Evaluate (4)2(-4)^2:
    (4)2=(4)×(4)=16(-4)^2 = (-4) \times (-4) = 16.
  • Now, evaluate 16216^2:
    162=16×16=25616^2 = 16 \times 16 = 256.

The value of ((4)2)2(( -4)^2)^2 is 256256.

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

  • Evaluate (4)2(4)^2:
    42=4×4=164^2 = 4 \times 4 = 16.
  • Now, consider the negative sign: (4)2=(16)=16-(4)^2 = -(16) = -16.
  • Evaluate (16)2(-16)^2:
    (16)2=(16)×(16)=256(-16)^2 = (-16) \times (-16) = 256.

The value of ((4)2)2(-(4)^2)^2 is 256256.

Therefore, the two expressions are equal.

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

Answer

= =

Exercise #10

Which is larger?

((3)3)2((2)2)4 (-(-3)^3)^2⬜((2)^2)^4

Video Solution

Step-by-Step Solution

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

  • Step 1: Evaluate the first expression ((3)3)2 (-(-3)^3)^2
  • Step 2: Evaluate the second expression ((2)2)4((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 (-(-3)^3)^2 .
The inner expression is (3)3(-3)^3. Calculating this gives: (3)3=27 (-3)^3 = -27 Next, we compute the expression (3)3-(-3)^3, which simplifies to: (27)=27 -(-27) = 27 Finally, we square this result: (27)2=729 (27)^2 = 729 Thus, the value of the first expression is 729.

Step 2: Evaluate the expression ((2)2)4((2)^2)^4.
First, calculate (2)2(2)^2: (2)2=4 (2)^2 = 4 Next, raise this result to the fourth power: (4)4=256 (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 (-(-3)^3)^2 = 729 and ((2)2)4=256((2)^2)^4 = 256 .

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

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

Answer

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