Powers of Negative Numbers Practice Problems and Solutions

Master negative number exponentiation with step-by-step practice problems. Learn even vs odd power rules, parentheses placement, and solve real examples.

📚Master Powers of Negative Numbers with Interactive Practice

Determine whether negative numbers raised to even powers give positive results
Calculate negative numbers raised to odd powers correctly
Distinguish between (-x)ⁿ and -xⁿ using parentheses rules
Apply exponent laws to solve complex negative number expressions
Identify common mistakes in negative number exponentiation
Build confidence solving multi-step problems with negative bases

Understanding Powers of Negative Numbers

Complete explanation with examples

Exponentiation of Negative Numbers

Negative number raised to an even power

Raising any negative number to an even power will result in a positive outcome.
When $n$ is even:
$(-x)^n=x^n$

Negative number raised to an odd power

Raising any negative number to an odd power will result in a negative outcome.
When $n$ is odd:
$(-x)^n=-(x)^n$

What is the difference between a power that is inside parentheses and one that is outside of them?

When the exponent is outside the parentheses - it applies to everything inside them.
When the exponent is inside the parentheses - it applies only to its base and not to the minus sign that precedes it.

Detailed explanation

Practice Powers of Negative Numbers

Test your knowledge with 17 quizzes

$-(-1)^{80}=$

Examples with solutions for Powers of Negative Numbers

Step-by-step solutions included

Exercise #1

$(-2)^7=$

Step-by-Step Solution

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 $-128$ .

Answer:

$-128$

Video Solution

Exercise #2

$-(2)^2=$

Step-by-Step Solution

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 $-4$ .

Answer:

$-4$

Video Solution

Exercise #3

Solve the following expression:

$(-8)^2=$

Step-by-Step Solution

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

Answer:

$64$

Video Solution

Exercise #4

$-(7)^2=$

Step-by-Step Solution

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 $-49$ .

Answer:

$-49$

Video Solution

Exercise #5

$36=$

Step-by-Step Solution

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 $(-6)^2$ .

Answer:

$(-6)^2$

Video Solution

Frequently Asked Questions

Everything you need to know about Powers of Negative Numbers

What happens when you raise a negative number to an even power?

When a negative number is raised to an even power, the result is always positive. For example, (-3)² = 9 because (-3) × (-3) = 9. This follows the rule that negative times negative equals positive.

Why do negative numbers raised to odd powers stay negative?

Negative numbers raised to odd powers remain negative because you multiply an odd number of negative values. For instance, (-2)³ = (-2) × (-2) × (-2) = -8, where the final multiplication involves one negative factor.

What's the difference between (-4)² and -4²?

(-4)² equals 16 because the exponent applies to the entire negative number in parentheses. However, -4² equals -16 because the exponent only applies to 4, then the negative sign is applied afterward.

How do parentheses affect negative number exponentiation?

Parentheses determine what the exponent applies to: • With parentheses: (-x)ⁿ means the exponent applies to the entire negative number • Without parentheses: -xⁿ means the exponent applies only to x, then add the negative sign

What are the most common mistakes with negative number powers?

Common errors include: 1) Forgetting that even powers of negatives are positive, 2) Misapplying exponents when parentheses are missing, 3) Confusing (-x)ⁿ with -xⁿ notation.

How do you solve complex expressions like (-2)⁴ - 3²?

Solve step by step: First, calculate (-2)⁴ = 16 (even power makes it positive). Then calculate 3² = 9. Finally, compute 16 - 9 = 7. Remember to follow order of operations.

When do negative base exponents appear in real life?

Negative base exponents appear in physics (alternating currents), finance (debt calculations), and temperature changes. Understanding these rules helps solve problems involving periodic patterns and directional quantities.

What grade level typically learns powers of negative numbers?

Powers of negative numbers are typically introduced in 8th grade algebra and reinforced in high school algebra courses. Students should first master basic exponent rules and integer operations before tackling this concept.

Continue Your Math Journey

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Topics Learned in Later Sections

Exponents - Special Cases Zero Exponent Rule Negative Exponents

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Powers of Negative Numbers Practice Problems and Solutions

Understanding Powers of Negative Numbers

Exponentiation of Negative Numbers

Negative number raised to an even power

Negative number raised to an odd power

What is the difference between a power that is inside parentheses and one that is outside of them?

Practice Powers of Negative Numbers

Examples with solutions for Powers of Negative Numbers

Frequently Asked Questions

What happens when you raise a negative number to an even power?

Why do negative numbers raised to odd powers stay negative?

What's the difference between (-4)² and -4²?

How do parentheses affect negative number exponentiation?

What are the most common mistakes with negative number powers?

How do you solve complex expressions like (-2)⁴ - 3²?

When do negative base exponents appear in real life?

What grade level typically learns powers of negative numbers?

More Powers of Negative Numbers Questions