Powers of Negative Numbers
Powers of Negative Numbers addresses the important distinction between (-5)² = 25 (a negative number inside parentheses raised to a power) and -5² = -25 (a negative sign applied to the result of 5²). Covered in Yoshiwara Elementary Algebra Chapter 5: Exponents and Roots, this nuance is a frequent source of errors for Grade 6 students. Understanding the role of parentheses in exponent notation is essential for accurate algebraic computation.
Key Concepts
Property To show that a negative number is raised to a power, we enclose the negative number in parentheses. For example, to indicate the square of $ 5$, we write $$( 5)^2 = ( 5)( 5) = 25$$ If the negative number is not enclosed in parentheses, the exponent applies only to the positive number. $$ 5^2 = (5 \cdot 5) = 25$$.
Examples To calculate $( 3)^4$, we multiply four factors of $ 3$: $( 3)( 3)( 3)( 3) = 81$.
The expression $ 3^4$ means the negative of $3^4$, so we calculate $3 \cdot 3 \cdot 3 \cdot 3 = 81$ and then apply the negative sign to get $ 81$.
Common Questions
What is (-5) squared?
(-5)² = (-5)(-5) = 25. When a negative number is inside parentheses and raised to a power, you multiply the negative number by itself, producing a positive result for even exponents.
What is the difference between (-5)² and -5²?
(-5)² means the square of negative 5, which is 25. -5² means the negative of 5 squared, which is -25. The parentheses make all the difference.
When does a negative number raised to a power give a positive result?
A negative number raised to an even power is positive. A negative number raised to an odd power is negative.
Where are powers of negative numbers in Yoshiwara Elementary Algebra?
This concept is in Chapter 5: Exponents and Roots of Yoshiwara Elementary Algebra.
Why do parentheses matter with negative exponents?
Parentheses indicate the entire negative number is the base. Without them, only the positive number is raised to the power and the negative sign is separate.