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Wize High School Algebra I Textbook (Common Core) > Foundations of Algebra

📝Review: Powers & Exponents

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Powers & Exponents

Powers can be used to show repeated multiplication of the same number.


This is read as "two to the power of three"

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Example
Represent the following as powers, then simplify them:

a) 3×3×3×3×4×4+2×2×23\times3\times3\times3\times4\times4+2\times2\times2

3×3×3×3⏟four of these×2×2⏟two of these+4×4×4⏟three of these\underbrace{3\times3\times3\times3}_{\colorFour{\text{four of these}}} \times \underbrace{2\times2}_{\colorThree{\text{two of these}}} + \underbrace{4\times4\times4}_{\textcolor{#8727EE}{\text{three of these}}}

=34×22+43=3^{\bm{\colorFour{4}}}\times2^{\bm{\colorThree{2}}}+4^{\bm{\colorFive{3}}}

=81×4+64=81\times4+64

=324+64=324+64

=388=388


b) (−5)(−5)(−5)(-5)(-5)(-5)

(−5)(−5)(−5)⏟three of these\underbrace{(-5)(-5)(-5)}_{\colorFour{\text{three of these}}}

=(−5)3=(-5)^{\bm{\colorFour{3}}}

=−125=-125


c) x×7x×x×3y×y×7x\times7 x\times x\times 3y\times y\times 7

Grouping all the same bases together:
x×x×x⏟three of theses×y×y⏟two of theses×3⏟one of these×7×7⏟two of these\underbrace{x\times x\times x}_{\colorFour{\text{three of theses}}}\times \underbrace{y\times y}_{\colorThree{\text{two of theses}}}\times \underbrace{3}_{\colorTwo{\text{one of these}}}\times \underbrace{7\times 7}_{\colorFive{\text{two of these}}}

=x3×y2×31×72=x^{\bm{\colorFour{3}}} \times y^{\bm{\colorThree{2}}} \times 3^{\bm{\textcolor{#FF1AB3}{1}}} \times 7^{\bm{\textcolor{#8727EE}{2}}}

=x3×y2×3×49=x^3\times y^2\times3\times49

=147x3y2=147x^3y^2


Wize Tip
When multiplying variables and numbers together, we usually don't have to write ×\times between them.

For example, these are all the same
  • 3×x×y3\times x\times y
  • 3xy3xy
  • 3(x)(y)3(x)(y)
  • 3(xy)3(xy)

Practice: Representing Powers

Match the following expressions with their simplified form.
A.
3xy(4y2)(3yx4)3xy(4y^2)(3yx^4)
B.
−2a(2a)−2xy-2a(2a)-2xy

C.
−2a(2a)(−2xy)-2a(2a)(-2xy)
32×4x5y43^2\times4x^5y^4
−22a2−2xy-2^2a^2-2xy
23a2xy2^3a^2xy
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Powers with Positive and Negative Bases

Recall the following multiplication and division patterns:

Multiplication       Division(+)(+)→(+)(+)(+)→(+)(−)(−)→(+)(−)(−)→(+)(+)(−)→(−)(+)(−)→(−)(−)(+)→(−)(−)(+)→(−)\begin{array}{|lc|lc|} \hline \text{Multiplication}&~~~~~~~&\text{Division}\\ \hline\\ (\bm{\colorFour{+}})(\bm{\colorFour{+}})\to(\bm{\colorFour{+}})&& \displaystyle\frac{(\bm{\colorFour{+}})}{(\bm{\colorFour{+}})}\to(\bm{\colorFour{+}})\\\\ (\bm{\colorFour{-}})(\bm{\colorFour{-}})\to(\bm{\colorFour{+}})&& \displaystyle\frac{(\bm{\colorFour{-}})}{(\bm{\colorFour{-}})}\to(\bm{\colorFour{+}})\\\\ (\bm{\colorFour{+}})(\bm{\colorFour{-}})\to(\bm{\colorFour{-}})&& \displaystyle\frac{(\bm{\colorFour{+}})}{(\bm{\colorFour{-}})}\to(\bm{\colorFour{-}})\\\\ (\bm{\colorFour{-}})(\bm{\colorFour{+}})\to(\bm{\colorFour{-}})&&\displaystyle\frac{(\bm{\colorFour{-}})}{(\bm{\colorFour{+}})}\to(\bm{\colorFour{-}})\\\\ \hline \end{array}

Using these patterns, we have some rules for power with positive and negative bases:

Powers(+) any exponent → (+)(−) even exponent → (+)(−)  odd exponent → (−)\begin{array}{|c|} \hline \text{Powers}\\ \hline\\ (\bm{\colorFour{+}})^{\text{ any exponent}}~\to~(\bm{\colorFour{+}})\\\\ (\bm{\colorFour{-}})^{\text{ even exponent}}~\to~(\bm{\colorFour{+}})\\\\ (\bm{\colorFour{-}})^{~\text{ odd exponent}}~\to~(\bm{\colorFour{-}})\\\\ \hline \end{array}

Practice: Powers with Negative and Positive Bases

Without doing any calculations, decide if the anwer will be positive or negative.

a) (−5)3(-5)^3

b) −53-5^3

c) −(−5)3-(-5)^3

d) (−5)4(-5)^4

e) −54-5^4

f) −(−5)4-(-5)^4

g) −23×(−5)3-2^3\times(-5)^3

Enter positive or negative in the space provided

Practice: Evaluating Powers

Evaluate the following.

Each question part will tell you whether a calculator is allowed or not.

Without using a calculator, evaluate (−2)3+(−2)4(-2)^3+(-2)^4.

Practice: Evaluating Powers of Rational Numbers

Without using a calculator, evaluate the following:

a) (54)2\left(\dfrac{5}{4}\right)^2

b) (32)2\left(\dfrac{3}{2}\right)^2

c) (−32)2\left(\dfrac{-3}{2}\right)^2

d) (−32)3\left(-\dfrac{3}{2}\right)^3

Bonus:

Instead of evaluating these powers as multiplication of numbers, can you think of a short-cut way of evaluating powers of rational numbers?