Exponents with Negative Bases

Exponents with Negative Bases exercise

• Understand the rule to determine the sign of a negative number with an exponent.

  Hints

  If you ever forget the rule and are unsure if an expression in exponential notation with a negative base is positive or negative, you can always expand the expression.

  For example:

  $(-3)^4 = (-3)(-3)(-3)(-3)$

  When you have very large exponents, this may not be efficient, so the rule will then be useful.

  • When the base is negative and you have an even exponent, the solution is positive.
  • When the base is negative and you have an odd exponent, the solution is negative.

  Solution

  Since $(-1)^{100}$ has an even exponent of $\bf{100}$, it is equal to a positive one. The rule states that when there is a negative base, and the exponent is even, the result is always positive.

  For this example, $(-1)^{101}$ has an odd exponent of $101$, it is equal to a negative one. The rule states that when there is a negative base, and the exponent is odd, the result is always negative.

  The solution to $(-1)^{200}$ is 1.

  The solution to $(-1)^{199}$ is -1.

• Determine the sign of an exponent with a negative base.

  Hints

  When there is a negative base, the exponent will determine whether the solution will be positive or negative.

  Even Exponent = Positive Solution

  Odd Exponent = Negative Solution

  Here is an example to help you:

  $(-2)^4$

  The exponent of $4$ is even which will make our solution positive.

  $(-2)^7$

  The exponent of $7$ is odd which will make our solution negative.

  Solution

  Positive Solutions $(-3)^2$ $(-11)^8$ $(-5)^{10}$ $(-1)^{12}$

  Negative Solutions $(-2)^3$ $(-9)^{19}$ $(-5)^5$ $(-10)^9$

• Applying your knowledge of exponential notation.

  Hints

  The rule states that when you have a negative base and an exponent follow these guidelines for determining the sign of the solution.

  If you are using a calculator to check your solution, be sure to include the parentheses sign.

  The expression can be expanded to $(-2)(-2)(-2)(-2)(-2)$.

  First multiply the first two $(-2)(-2)=4$ and then continue with the remaining $(-2)$s.

  Solution

  $(-2)^5 = \bf{-32}$

  This expression can be expanded: $(-2)(-2)(-2)(-2)(-2)$.

  Since this expression with a negative base, has an odd exponent, the solution must be negative.

  The solution can be found by multiplying $-2$ by itself $5$ times.

• Identify characteristics in exponential notation that can determine a solution's sign.

  Hints

  Using a set of parentheses is super important when it comes to exponents.

  $(-1)^8$ and $-1^8$ are not the same thing.

  $(-1)^8$ expanded: $(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)$

  $-1^8$ expanded: $- (1)(1)(1)(1)(1)(1)(1)(1)$

  $(-1)^8 = 1$

  $-1^8 = -1$

  Solution

  Here are the matches.

  $(-3)^2=9$

  $-3^2=-9$

  $(-1)^8=1$

  $(-6)^1=-6$

  $-1^8=-1$

  $(-6)^2=36$

• Understand what it means to expand an expression in exponential notation.

  Hints

  Numbers written in exponential notation are made up of a base and an exponent.

  In $(-1)^7$, $\bf{(-1)}$ is the base and $\bf{7}$ is the exponent

  Numbers written in exponential notation can be expanded.

  In $(-1)^7$, we are multiplying $\bf{(-1)}$ by itself seven times.

  Solution

  The base is $(-1)$ and the exponent is $7$. This means that the $(-1)$ is multiplied by itself $7$ times and can be expanded like this:

  $\bf{(-1)(-1)(-1)(-1)(-1)(-1)(-1)}$

• Compare quantities in exponential notation with negative bases.

  Hints

  Examples of Inequality Symbols:

  4 > -1

  -3 < 8

  -5 = -5

  Use what you have learned about negative bases with an exponent.

  Take a look at this example to help you!

  $(-2)^4$ ? $(-5)^5$

  ${}$

  $(-2)^4$ > $(-5)^5$

  ${}$

  $16$ > $-3,125$

  Solution

  $(-2)^3$ < $(-2)^1$

  • $-8$ < $-2$

  ${}$

  $(-3)^2$ > $(-3)^3$

  • $9$ > $-27$

  ${}$

  $(-x)^2$ > $(-x)^9$

  • "a positive value" > "a negative value"

  ${}$

  $(-1)^4$ = $(-1)^8$

  • $1$ = $1$