exponent laws: an investigation
DESCRIPTION
Exponent Laws: An Investigation. 1 st Law : Product of Powers. Write the expanded form of the first product in the table. 1 st Law : Product of Powers. Write the expanded form of the first product in the table. 1 st Law : Product of Powers. Now write it as a single power. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/1.jpg)
Exponent Laws:An Investigation
![Page 2: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/2.jpg)
Write the expanded form of the first product in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24
52 x 51
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 3: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/3.jpg)
Write the expanded form of the first product in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2
52 x 51
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 4: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/4.jpg)
Now write it as a single power.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2
52 x 51
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 5: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/5.jpg)
Now write it as a single power.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 6: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/6.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 7: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/7.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 8: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/8.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3
p5 x p2
g3 x g x g4
![Page 9: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/9.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3 3x3 x 3x3x3 x 3
p5 x p2
g3 x g x g4
![Page 10: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/10.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3 3x3 x 3x3x3 x 3 36
p5 x p2
g3 x g x g4
![Page 11: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/11.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3 3x3 x 3x3x3 x 3 36
p5 x p2 pxpxpxpxp x pxp
g3 x g x g4
![Page 12: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/12.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3 3x3 x 3x3x3 x 3 36
p5 x p2 pxpxpxpxp x pxp p7
g3 x g x g4
![Page 13: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/13.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3 3x3 x 3x3x3 x 3 36
p5 x p2 pxpxpxpxp x pxp p7
g3 x g x g4 gxgxg x g x gxgxgxg
![Page 14: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/14.jpg)
Repeat the previous two instructions for the remaining products in the table.
1st Law: Product of Powers
Product Expanded Form Single Power
23 x 24 2x2x2 x 2x2x2x2 27
52 x 51 5x5 x 5 53
32 x 33 x 3 3x3 x 3x3x3 x 3 36
p5 x p2 pxpxpxpxp x pxp p7
g3 x g x g4 gxgxg x g x gxgxgxg g8
![Page 15: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/15.jpg)
What pattern can be seen?
![Page 16: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/16.jpg)
In General:When multiplying powers with the same base, keep the base and add the exponents.
y a y b y ab
![Page 17: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/17.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
![Page 18: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/18.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
![Page 19: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/19.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
![Page 20: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/20.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
![Page 21: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/21.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
![Page 22: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/22.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
99999999999
![Page 23: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/23.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
99999999999
91
![Page 24: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/24.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
99999999999
91
ppppppp
![Page 25: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/25.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
99999999999
91
ppppppp
p3
![Page 26: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/26.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
99999999999
91
ppppppp
p3
gggg
![Page 27: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/27.jpg)
2nd Law: Quotient of Powers
Quotient Expanded Form Single Power
85 83
64 61
96 95
p5 p2
g3 g
For each quotient, write both its parts in expanded form and then as a single power.
88888888
82
66666
63
99999999999
91
ppppppp
p3
gggg
g2
![Page 28: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/28.jpg)
What pattern can be seen?
![Page 29: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/29.jpg)
In General:When dividing powers with the same base, keep the base and subtract the exponents.
y a y b y a b
![Page 30: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/30.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
![Page 31: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/31.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
43 43 444444
![Page 32: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/32.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
43 43 444444
46
![Page 33: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/33.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
43 43 444444
46
72 72 72 72 77777777
![Page 34: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/34.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
43 43 444444
46
72 72 72 72 77777777
78
![Page 35: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/35.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
43 43 444444
46
72 72 72 72 77777777
78
y 4 y 4 y 4 yyyyyyyyyyyy
![Page 36: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/36.jpg)
3rd Law: Powers of Powers
Power of a Power Expanded Form Single Power
43 2
72 4
y 4 3
Write each power of a power in expanded form and then as a single power.
43 43 444444
46
72 72 72 72 77777777
78
y 4 y 4 y 4 yyyyyyyyyyyy
y12
![Page 37: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/37.jpg)
What pattern can be seen?
![Page 38: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/38.jpg)
In General:When writing a power of a power as a single power, keep the base and multiply the exponents.
y a b y ab
![Page 39: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/39.jpg)
Try a few examples…
![Page 40: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/40.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
![Page 41: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/41.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
36
![Page 42: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/42.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
36
729
![Page 43: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/43.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
36
729
53
![Page 44: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/44.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
36
729
53
125
![Page 45: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/45.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
36
729
53
125
29
![Page 46: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/46.jpg)
Write each as a single power, then evaluate using a calculator.
34 32
58 55
23 3a) b) c)
36
729
53
125
29
512
![Page 47: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/47.jpg)
Now try these…
![Page 48: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/48.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
![Page 49: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/49.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
x 345
![Page 50: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/50.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
x 345
x12
![Page 51: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/51.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
x 345
x12
r19 8
![Page 52: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/52.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
x 345
x12
r19 8
r11
![Page 53: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/53.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
x 345
x12
r19 8
r11
b123
![Page 54: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/54.jpg)
Simplify.
x 3 x 4 x 5
r19 r8
b12 3a) b) c)
x 345
x12
r19 8
r11
b123
b36
![Page 55: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/55.jpg)
Now try a few tricky ones…
![Page 56: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/56.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
![Page 57: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/57.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
![Page 58: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/58.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
![Page 59: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/59.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
![Page 60: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/60.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
12x 7y 2
2xy 2xy 2xy xy 3
3x
![Page 61: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/61.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
12x 7y 2
2xy 2xy 2xy xy 3
3x
12x 7y 2
2 2 2 x x x y y y xy 3
3x
![Page 62: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/62.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
12x 7y 2
2xy 2xy 2xy xy 3
3x
12x 7y 2
2 2 2 x x x y y y xy 3
3x
12x 7y 2
8x 3y 3
xy 3
3x
![Page 63: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/63.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
12x 7y 2
2xy 2xy 2xy xy 3
3x
12x 7y 2
2 2 2 x x x y y y xy 3
3x
12x 7y 2
8x 3y 3
xy 3
3x
12 1 x 7 x y 2 y 3
8 3 x 3 x y 3
![Page 64: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/64.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
12x 7y 2
2xy 2xy 2xy xy 3
3x
12x 7y 2
2 2 2 x x x y y y xy 3
3x
12x 7y 2
8x 3y 3
xy 3
3x
12 1 x 7 x y 2 y 3
8 3 x 3 x y 3
12x 8y 5
24x 4y 3
![Page 65: Exponent Laws: An Investigation](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815792550346895dc525f8/html5/thumbnails/65.jpg)
Simplify.
a) b)
xy 3 2x 5y 6
12x 7y 2
2xy 3
xy 3
3x
1xy 3 2x 5y 6
12x1 x 5 y 3 y 6
2x 6y 9
12x 7y 2
2xy 2xy 2xy xy 3
3x
12x 7y 2
2 2 2 x x x y y y xy 3
3x
12x 7y 2
8x 3y 3
xy 3
3x
12 1 x 7 x y 2 y 3
8 3 x 3 x y 3
12x 8y 5
24x 4y 3
x 4y 2
2