HSPT: Comparisons: negatives and exponents
Learn to evaluate expressions that mix negative signs and exponents, and to order negative numbers correctly.
How do you compare two quantities when negative signs and exponents are mixed together?
Short answer
Evaluate each quantity, paying attention to whether the negative is inside the parentheses. Then compare on a number line: farther right is greater.
Know first: Exponents, Number lines with negative numbers
Step 1 of 5
Where the negative sign sits changes the value
A negative sign and an exponent can combine in different ways. (−8)² squares negative 8: (−8) × (−8) = 64. −(8²) squares 8 first, then makes it negative: −64. Without parentheses, −8² means −(8²), so it is also −64.
Once you have the values, compare them on a number line. Farther right is greater, so a negative number closer to zero is greater.
| Expression | What happens | Value |
|---|---|---|
| (−8)² | Square negative 8 | 64 |
| −(8²) | Square 8, then make it negative | −64 |
| −8² | Same as −(8²) | −64 |
| −(−8) | The opposite of negative 8 | 8 |
Step 2 of 5
Evaluate, then place on a number line
Worked example: A square made negative
Compare Quantity A: −(9²) and Quantity B: −80.
- AQuantity A is greater.
- BQuantity B is greater.
- CThe two quantities are equal.
- DThe relationship cannot be determined.
- 1
Evaluate A
9² = 81, so −(9²) = −81.
- 2
Place both values
−81 is one step farther left than −80.
- 3
Compare
The value farther right is greater, so −80 is greater.
- 4
See why the other choices are there
A looks greater if you compare 81 and 80 and forget the signs. Equal would need A to be −80. Cannot be determined doesn't fit, since both quantities are fixed numbers.
Answer
Quantity B is greater.
Find the wrong step
A student compares Quantity A: (−2)³ and Quantity B: −(2²).
- 1
Finds A: (−2) × (−2) × (−2) = −8.
- 2
Finds B: 2² = 4, so B is −4.
- 3
Decides A is greater because 8 is bigger than 4.
What went wrong
For negative numbers, a bigger distance from zero means a smaller value. −8 is to the left of −4.
The fix
−4 is closer to zero than −8, so Quantity B is greater.
Step 3 of 5
Watch the sign and the parentheses
Check yourself · Question 1
Compare Quantity A: −(10²) and Quantity B: −99.
Answer: B
A = −(100) = −100. On a number line, −100 is to the left of −99, so −99 is greater.
Trap answer: Quantity A is greater.
Comparing without the signs
- Why it looks right
- 100 is bigger than 99, and that fact jumps out before the signs register.
- Why it's wrong
- Both numbers are negative. −100 is farther from zero on the left side, so it is smaller.
- The right answer
- Quantity B is greater, because −99 is to the right of −100.
Check yourself · Question 2
Compare Quantity A: −(5²) + 61 and Quantity B: (−6)².
Answer: D
A: 5² = 25, so −25 + 61 = 36. B: (−6) × (−6) = 36. The quantities are equal.
Common mistake
"(−6)² and −6² are the same thing."
- Why it's tempting
- They use the same digits and the same signs.
- Do this instead
- The parentheses decide what gets squared. (−6)² = 36, but −6² = −36. Write out the multiplication if you're unsure.
Step 4 of 5
What to remember
Remember
- (−n)² is positive; −(n²) and −n² are negative.
- Evaluate each quantity fully before you compare.
- Among negative numbers, the one closer to zero is greater.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What was the value of each quantity in the practice question, and which one sits farther right on a number line?