Product & Quotient Rules of Exponents

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8th grade

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Mathematics

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Slide 1
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Product & Quotient Rules of Exponents
Laws of Exponents
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Slide 2
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Product Rule
Look at the right and see what happens when you multiply powers that have the same base
When you expand 3² you have two 3s being multiplied and when you expand 3³ you have three 3s being multiplied.
All together how many 3s are you multiplying?
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Slide 3
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Product Rule Short Cut?!!
When you multiply same base powers you keep the base and add the exponents.
x² * x⁷= x²⁺⁷ = x⁹
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Question 4
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Q.
Simplify the following:
$a^{11}\cdot a^2\cdot a^0$
answer choices
$a^{13}$
$1$
$a^{22}$
$a^0$
 <math><semantics><mrow><msup><mi>a</mi><mn>13</mn></msup></mrow><annotation encoding="application/x-tex">a^{13}</annotation></semantics></math>a13 
alternatives
 <math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>1 
 <math><semantics><mrow><msup><mi>a</mi><mn>22</mn></msup></mrow><annotation encoding="application/x-tex">a^{22}</annotation></semantics></math>a22 
 <math><semantics><mrow><msup><mi>a</mi><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">a^0</annotation></semantics></math>a0 
answer explanation
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Question 5
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Q.
Simplify the following:
$x^3\cdot y^2\cdot x^2$
answer choices
$xy^7$
$x^5y^2$
$\left(xy\right)^7$
$x^5$
 <math><semantics><mrow><mi>x</mi><msup><mi>y</mi><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">xy^7</annotation></semantics></math>xy7 
alternatives
 <math><semantics><mrow><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^5y^2</annotation></semantics></math>x5y2 
 <math><semantics><mrow><msup><mrow><mo fence="true">(</mo><mi>x</mi><mi>y</mi><mo fence="true">)</mo></mrow><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\left(xy\right)^7</annotation></semantics></math>(xy)7 
 <math><semantics><mrow><msup><mi>x</mi><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">x^5</annotation></semantics></math>x5 
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Question 6
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Q.
Simplify the following:
$x^3\cdot x\cdot x^{-2}$
answer choices
$x^1$
$x^5$
$x^6$
$x^2$
 <math><semantics><mrow><msup><mi>x</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">x^1</annotation></semantics></math>x1 
alternatives
 <math><semantics><mrow><msup><mi>x</mi><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">x^5</annotation></semantics></math>x5 
 <math><semantics><mrow><msup><mi>x</mi><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">x^6</annotation></semantics></math>x6 
 <math><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2</annotation></semantics></math>x2 
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Slide 7
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Quotient Rule
When dividing powers that have the same base think of dividing as the inverse of multiplying, so what is the inverse of adding?
When you expand x⁵ you have five Xs being multiplied in the numerator and when you expand x² you have two Xs being multiplied in the denominator.
So what happens when you have the Xs in the numerator and Xs in the denominator? Yup! you cross out the matching pairs!
Now what did you end up with?
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Slide 8
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Quotient Rule Short Cut?!!
When you divide same base powers you keep the base and subtract the exponents.
x⁷ / x²= x^7-2 = x⁵
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Question 9
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Q.
Simplify the following:
$\frac{x^{16}}{x^2}$
answer choices
$x^{14}$
$x^8$
$x^{18}$
$x^{-8}$
 <math><semantics><mrow><msup><mi>x</mi><mn>14</mn></msup></mrow><annotation encoding="application/x-tex">x^{14}</annotation></semantics></math>x14 
alternatives
 <math><semantics><mrow><msup><mi>x</mi><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">x^8</annotation></semantics></math>x8 
 <math><semantics><mrow><msup><mi>x</mi><mn>18</mn></msup></mrow><annotation encoding="application/x-tex">x^{18}</annotation></semantics></math>x18 
 <math><semantics><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>8</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{-8}</annotation></semantics></math>x−8 
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Question 10
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Q.
Simplify the following:
$\frac{x^9y^7}{x^2y^6}$
answer choices
$xy^8$
$\left(xy\right)^{24}$
$x^7y$
$x^{11}y^1$
 <math><semantics><mrow><mi>x</mi><msup><mi>y</mi><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">xy^8</annotation></semantics></math>xy8 
alternatives
 <math><semantics><mrow><msup><mrow><mo fence="true">(</mo><mi>x</mi><mi>y</mi><mo fence="true">)</mo></mrow><mn>24</mn></msup></mrow><annotation encoding="application/x-tex">\left(xy\right)^{24}</annotation></semantics></math>(xy)24 
 <math><semantics><mrow><msup><mi>x</mi><mn>7</mn></msup><mi>y</mi></mrow><annotation encoding="application/x-tex">x^7y</annotation></semantics></math>x7y 
 <math><semantics><mrow><msup><mi>x</mi><mn>11</mn></msup><msup><mi>y</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">x^{11}y^1</annotation></semantics></math>x11y1 
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Question 11
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Q.
Simplify the following:
$\frac{x^7}{x^{-4}}$
answer choices
$x^{11}$
$x^3$
$x^{-3}$
$x^{-11}$
 <math><semantics><mrow><msup><mi>x</mi><mn>11</mn></msup></mrow><annotation encoding="application/x-tex">x^{11}</annotation></semantics></math>x11 
alternatives
 <math><semantics><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">x^3</annotation></semantics></math>x3 
 <math><semantics><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{-3}</annotation></semantics></math>x−3 
 <math><semantics><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{-11}</annotation></semantics></math>x−11 
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Question 12
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Q.
Simplify the following:
$\frac{x^3}{x^8}$
answer choices
$x^{-5}$
$x^5$
$\frac{1}{x^5}$
$\frac{1}{x^{11}}$
 <math><semantics><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{-5}</annotation></semantics></math>x−5 
alternatives
 <math><semantics><mrow><msup><mi>x</mi><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">x^5</annotation></semantics></math>x5 
 <math><semantics><annotation encoding="application/x-tex">\frac{1}{x^5}</annotation></semantics></math>x51 
 <math><semantics><mrow><mfrac><mn>1</mn><msup><mi>x</mi><mn>11</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{x^{11}}</annotation></semantics></math>x111 
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Question 13
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