Powers of Negative Numbers
Grade 8 math lesson on powers of negative numbers and determining when the result is positive or negative. Students learn the rule that negative numbers raised to even powers are positive and to odd powers are negative, and apply this to simplifying expressions.
Key Concepts
Property Squaring a negative number results in a positive number. Cubing a negative number results in a negative number. An even number of negative factors is positive, and an odd number of negative factors is negative.
Examples $( 3)^2 = ( 3)( 3) = 9$ $( 3)^3 = ( 3)( 3)( 3) = 27$ $( 1)^{10} = 1$.
Explanation When you raise a negative number to a power, it’s all about pairing up the negative signs. If the exponent is an even number, every negative sign finds a partner and they turn positive together. But if the exponent is odd, one lonely negative sign is left over, making the final answer negative every single time.
Common Questions
What is a negative number raised to a power?
A negative number raised to a power means multiplying the negative number by itself the given number of times. The sign of the result depends on whether the exponent is even or odd.
Why is a negative number raised to an even power positive?
Negative times negative is positive. So (-2)^4 = (-2)(-2)(-2)(-2) = (+4)(+4) = 16. Any negative number raised to an even power results in a positive number because the negatives pair up.
Why is a negative number raised to an odd power negative?
After pairing up negatives to get positives, one negative factor is left over. So (-2)^3 = (-2)(-2)(-2) = (+4)(-2) = -8. An odd power means one unpaired negative remains, making the result negative.
What is the difference between (-2)^4 and -2^4?
These are different. (-2)^4 = (-2)(-2)(-2)(-2) = 16 because the negative is included in the base. But -2^4 means -(2^4) = -(16) = -16 because only 2 is raised to the power, then negated.