9G (2.10) Operation with Exponents

9G (2.10)
Operations with Exponent

Distributing an exponent across parentheses.

Remember: It can be very helpful to use variable when we want to talk about "any number".

Multiple Choice

$$\left(4x\right)^2=?$$

[Square each thing in the parentheses by itself]

• $$4^2\cdot x^2=16x^2$$

• 8x

Multiple Choice

$$\left(2x\right)^2=?$$

[Square each thing in the parentheses by itself]

• $$2^2\cdot x^2=4x^2$$

• 2x

Multiple Choice

$$\left(2x\right)^3=?$$

• $$4x^2$$

• $$8x^3$$

Multiple Choice

$$\left(7x\right)^2=?$$

• $$49x^2$$

• $$8x^3$$

Multiple Choice

$$\left(7x\right)^3=?$$

• $$343x^3$$

• $$8x^3$$

Multiple Choice

$$\left(3x\right)^3=?$$

• $$27x^3$$

• $$8x^3$$

• $$9x^3$$

• $$27x^2$$

• 3x

Multiple Choice

$$\left(\frac{x}{3}\right)^3=?$$

• $$\frac{x^3}{3^3}\ \ \ =\ \ \ \frac{x^3}{27}$$

• $$27x^3$$

Multiple Choice

$$\left(\frac{x}{2}\right)^3=?$$

• $$\frac{x^3}{9}$$

• $$8x^3$$

• $$\frac{x^3}{8}$$

• 3x

Multiple Choice

$$\left(\frac{x}{3}\right)^4=?$$

• $$\frac{x^3}{27}$$

• $$\frac{x^3}{81}$$

• $$9x^3$$

• $$\frac{x^3}{8}$$

• 3x

Multiple Choice

$$\left(\frac{x}{7}\right)^2=?$$

• $$\frac{x^2}{49}$$

• $$8x^3$$

• $$9x^3$$

• $$\frac{x^2}{14}$$

• 3x

Multiple Choice

$$\left(x^1\right)^2=?$$

[Try to write it out]

• $$x^1\cdot x^1\ \ =x^2$$

• $$x^3$$

Multiple Choice

$$\left(x^3\right)^2=?$$

[Try to write it out]

• $$x^3\cdot x^3=\ \ \ x\cdot x\cdot x\cdot x\cdot x\cdot x\ \ \ \ =\ x^6$$

• $$x^3$$

Multiple Choice

$$\left(x^2\right)^4=?$$

[Try to write it out]

• $$x^2\cdot x^2\cdot x^2\cdot x^2=\ \ \ x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\ \ \ \ =\ x^8$$

• $$x^6$$

Multiple Choice

$$\left(x^5\right)^2=?$$

[Another way to remember is exponent outside the parantheses multiplies

with exponents on the inside ]

• $$2\cdot5\ =\ 10\ \ \ \ \ so\ \ x^{10}$$

• $$2\cdot5\ =\ 6\ \ \ \ \ so\ x^6$$

• $$2\cdot5\ =\ 7\ \ \ \ so\ \ \ x^7$$

Multiple Choice

$$\left(x^4\right)^3=?$$

[Another way to remember is exponent outside the parantheses multiplies

with exponents on the inside ]

• $$4\cdot3\ =\ 12\ \ \ \ \ so\ \ x^{12}$$

• $$4\cdot3\ =\ 10\ \ \ \ \ so\ x^{10}$$

• $$4\cdot3\ =\ 7\ \ \ \ so\ \ \ x^7$$

• $$4+3\ =\ 7\ \ \ \ so\ \ \ x^7$$

Multiple Choice

$$\left(\frac{x^2}{2}\right)^3=?$$

• $$\frac{\left(x^2\right)^3}{2^3}=\frac{x^6}{8}$$

• $$\frac{x^5}{8}$$

Multiple Choice

$$\left(\frac{x^3}{4}\right)^3=?$$

• $$\frac{x^9}{64}$$

• $$\frac{x^6}{64}$$

Multiple Choice

$$\left(\frac{x^4}{4}\right)^3=?$$

• $$\frac{x^{12}}{64}$$

• $$\frac{x^7}{64}$$

Multiple Choice

$$\left(x^3\right)^3=?$$

• $$\frac{x^9}{64}$$

• $$\frac{x^7}{64}$$

• $$\frac{x^6}{6}$$

• $$x^9$$

• $$x^6$$

Multiple Choice

$$\left(\frac{x^5}{2}\right)^3=?$$

• $$\frac{x^8}{8}$$

• $$8x^{15}$$

• $$\frac{x^{15}}{8}$$

• $$\frac{x^2}{8}$$

• $$\frac{x^5}{8}$$

Adding and Subtracting exponents.

Multiple Choice

$$x^2+x^2\ =\ ?$$

• $$1x^2+1x^2=2x^2$$

• $$x^4$$

Multiple Choice

$$x^2+2x^2\ =\ ?$$

• $$2x^2$$

• $$3x^2$$

• $$4x^2$$

Multiple Choice

$$2x^2+2x^2\ =\ ?$$

• $$2x^2$$

• $$3x^2$$

• $$4x^2$$

• $$5x^2$$

• $$6x^2$$

Multiple Choice

$$2x^2+3x^2\ =\ ?$$

• $$2x^2$$

• $$3x^2$$

• $$4x^2$$

• $$5x^2$$

• $$6x^2$$

Multiple Choice

$$3x^2+3x^2\ =\ ?$$

• $$2x^2$$

• $$3x^2$$

• $$4x^2$$

• $$5x^2$$

• $$6x^2$$

Multiple Choice

$$17x^2-5x^2\ =\ ?$$

• $$17x^2$$

• $$5x^2$$

• $$7x^2$$

• $$11x^2$$

• $$12x^2$$

Multiple Choice

$$17^2-5^2\ =\ ?$$

• $$17x^2$$

• 264

• $$7x^2$$

• $$11x^2$$

• $$12x^2$$

Multiple Choice

$$3x^3-2x^3\ =\ ?$$

• $$x^3$$

• $$2x^3$$

• $$7x^3$$

• $$5x^3$$

• $$x^2$$

Multiple Choice

$$2x^3+2x^3\ =\ ?$$

• $$x^3$$

• $$2x^3$$

• $$4x^3$$

• $$5x^3$$

• $$x^2$$

Multiple Choice

$$x+x^2\ =\ ?$$

• $$x^3$$

• $$2x^{ }$$

• $$2x^3$$

• $$2x^2$$

• $$x+x^2$$

You can only add them when they have the same exponent

Multiple Choice

$$2x^2+2x^3\ =\ ?$$

• $$x^3$$

• $$2x^3$$

• $$4x^3$$

• $$5x^3$$

• $$2x^2+2x^3$$

Multiple Choice

$$4^2+4^3\ =\ ?$$

• $$80$$

• $$4^5$$

• $$2\left(4^2\right)$$

• $$5x^3$$

• $$2x^2+2x^3$$

multiplying and dividing exponents

Multiple Choice

$$x^1\left(x^2\right)=?$$

• $$x^3$$

• $$x^2$$

Multiple Choice

$$x^2\left(x^3\right)=?$$

• $$x^5$$

• $$x^6$$

Multiple Choice

$$x^3\left(x^3\right)=?$$

• $$x^9$$

• $$x^4$$

• $$2x^2$$

• $$x^6$$

• $$2x^3$$

Multiple Choice

$$x^4\left(x^4\right)=?$$

• $$x^8$$

• $$2x^4$$

• $$x^{16}$$

• $$x^6$$

• $$2x^3$$

Multiple Choice

$$x^{12}\left(x^8\right)=?$$

• $$x^{96}$$

• $$x^{20}$$

• $$x^4$$

• $$2x^{12}$$

• $$2x^3$$

Multiple Choice

$$x^{10}\left(x^{13}\right)=?$$

• $$2x^{10}$$

• $$2x^{13}$$

• $$x^3$$

• $$x^{23}$$

• $$x^{120}$$

Let's talk about dividing by an exponent

Multiple Choice

$$\frac{x^2}{x}=?$$

• $$x$$

• $$0$$

Multiple Choice

$$\frac{x^3}{x}=?$$

• $$x^2$$

• $$x$$

Multiple Choice

$$\frac{x^4}{x}=?$$

• $$x^3$$

• $$x^2$$

Multiple Choice

$$\frac{x^4}{x^2}=?$$

• $$x^2$$

• $$x^3$$

Multiple Choice

$$\frac{x^5}{x^3}=?$$

• $$x^2$$

• $$x^3$$

Multiple Choice

$$\frac{x^5}{x}=?$$

• $$x^6$$

• $$x^4$$

• $$x^3$$

• $$2x^4$$

• $$2x^3$$

Multiple Choice

$$\frac{2x^3}{2x}=?$$

• $$x^2$$

• $$x^4$$

• $$x^3$$

• $$2x^4$$

• $$2x^2$$

Multiple Choice

$$\frac{2x^5}{x}=?$$

• $$x^6$$

• $$x^4$$

• $$x^3$$

• $$2x^4$$

• $$2x^3$$

Multiple Choice

$$\frac{x^5}{2x}=?$$

• $$x^6$$

• $$\frac{x^4}{2}$$

• $$x^3$$

• $$2x^4$$

• $$2x^3$$

Multiple Choice

$$\frac{4x^5}{2x^2}=?$$

• $$2x^7$$

• $$2x^3$$

• $$x^3$$

• $$2x^4$$

Multiple Choice

$$\frac{x^5}{x^4}=?$$

• $$2x^7$$

• $$2x^3$$

• $$x^3$$

• $$2x^4$$

• $$x$$

Multiple Choice

$$x^{-2}=\ ?$$

• $$x^2$$

• $$\frac{1}{x^2}$$

Multiple Choice

$$x^{-3}=\ ?$$

• $$x^3$$

• $$\frac{1}{x^3}$$

Multiple Choice

$$\frac{1}{x^{-2}}=\ ?$$

• $$\frac{1}{x^2}$$

• $$x^2$$

Multiple Choice

$$\frac{x^5}{x^1}\ is\ the\ same\ as\ \ x^5\left(x^{-1}\right)$$

$$x^5\left(x^{-1}\right)=?$$

• $$5-1\ =\ 4\ \ \ \ \ so\ x^4$$

• $$x^{-5}$$

Multiple Choice

$$\frac{x^3}{x^1}\ is\ the\ same\ as\ \ x^3\left(x^{-1}\right)$$

$$x^3\left(x^{-1}\right)=?$$

• $$3-1=2\ \ \ \ \ \ \ so\ x^2$$

• $$x^3$$

Multiple Choice

$$x^2\left(x^{-2}\right)=?$$

• x

• $$2x^3$$

• $$x^3$$

• $$2x^4$$

• 1

Multiple Choice

$$x^6\left(x^{-3}\right)=?$$

• $$2x^7$$

• $$2x^3$$

• $$x^3$$

• $$2x^4$$

• $$x$$

Multiple Choice

$$x^{10}\left(x^{-4}\right)=?$$

• $$2x^7$$

• $$2x^3$$

• $$x^3$$

• $$2x^4$$

• $$x^6$$

Multiple Choice

$$x^3\left(x^{-5}\right)=?$$

• $$2x^{-5}$$

• $$2x^3$$

• $$x^2$$

• $$2x^4$$

• $$x^{-2}$$

Multiple Choice

$$x^4\left(x^{-8}\right)=?$$

• $$x^{-4}$$

• $$2x^3$$

• $$x^2$$

• $$2x^4$$

• $$x^{-2}$$

Multiple Choice

$$\left(x^1\right)^{-2}=?$$

• $$x^{-1}$$

• $$1\left(-2\right)=-2\ \ \ \ \ \ \ \ so\ x^{-2}$$

Radicals by themselves are often irrational, meaning their digits go on forever, so we can't even write down without cutting off digits.


So it's not so useful to add two irrational numbers togethor

Multiple Choice

$$\sqrt[2]{x}+\sqrt[2]{x}\ =\ ?$$

• $$2\left(\sqrt[2]{x}\right)$$

• $$\sqrt[2]{x}$$

multiplying by radicals

Multiple Choice

$$\sqrt[2]{x}\left(\sqrt[2]{x}\right)=\ ?$$

• $$\left(\sqrt[2]{x}\right)^2=x$$

• 0

Multiple Choice

$$x^{\frac{1}{2}}\left(x^{\frac{1}{2}}\right)\ =?$$

[Just add the exponents like before]

• $$\frac{1}{2}+\frac{1}{2}=\frac{2}{2}=1\ \ \ \ \ so\ x^1$$

• $$x^{\frac{1}{4}}$$

Multiple Choice

$$-x^{\frac{1}{2}}\left(-x^{\frac{1}{2}}\right)\ =?$$

• $$\frac{1}{2}+\frac{1}{2}=\frac{2}{2}$$

  and negative times a negative is a positive

  so

  $$x^{\frac{2}{2}}=x^1$$

• 0

Multiple Choice

$$x^{\frac{1}{2}}\left(x^{-\frac{1}{2}}\right)=\ ?$$

• 1

  because

  $$\frac{1}{2}-\frac{1}{2}=0\ \ \ so\ x^0$$

• 0

  because

  $$\frac{1}{2}-\frac{1}{2}=0\ \ \ so\ x^0$$

THE END