Geometry Integrated with Algebra

1
Geometry Integrated with Algebra

• Section 2-6
• Using Powers

2
Focus

• Use negative, zero, and fractional exponents

3
Exploration

• What definition of negative and zero exponents
  does your calculator use?
• Material scientific or graphics calculator
• Work with another student

4
Exploration 1

• a. To complete the second column of Table 1, use
  a calculator to find the value of the expression
  in the first column.
• b. Use your calculator to show that the decimals
  you wrote in the second column are equal to the
  fraction in the third column (you can use the
  MATH key)
• c. What definition of x-n do you think your
  calculator uses? Check that this definition works
  for 8-1, 3-2,and (-5)-3

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Exploration 1Table 1
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Solution Exploration 1Table 1

• c. What definition of x-n do you think your
  calculator uses? Check that this definition works
  for 8-1, 3-2,and (-5)-3

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Exploration 2

• A. To complete Table 2, use a calculator to find
  the value of the expression in the first column.
• B. What definition of x0 do you think your
  calculator uses? Check that your calculator also
  uses this definition for negative values of x.

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Exploration 2Table 2
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Solution Exploration 2Table 2

• B. What definition of x0 do you think your
  calculator uses? Check that your calculator also
  uses this definition for negative values of x.
• x0 1 (-1)01 (-2)0 1 (-3)0 1

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Exploration 3

• To see if your definitions for x-n and x0 work
  when x0, use your calculator to evaluate 0-1,
  0-2, and 00. What happens? Why do you think this
  happens?

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Solution Exploration 3

• To see if your definitions for x-n and x0 work
  when x0, use your calculator to evaluate 0-1,
  0-2, and 00. What happens? Why do you think this
  happens?

The calculator gives an error message when the
suggested values are entered.
Since x-n is defined as , x-n is not defined
when xn 0 0n 0 for all positive numbers, n.
00 is not defined. (you can't divide by zero!)
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Definitions
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Sample 1

• Simplify 6b0a-4. Write the answer with positive
  exponents.

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Solution Sample 1

• Simplify 6b0a-4. Write the answer with positive
  exponents.
• Use the zero exponent rule 6b0a-4 61a-4
• Use the negative exponent rule6a-46

15
Exploration

• What Definition of x1/2 and x1/3 does your
  calculator use?
• Materials scientific or graphics calculators
• Work with another student

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Exploration 4

• Use a calculator to complete the table.

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Solution Exploration 4

• Use a calculator to complete the table.

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Exploration 5

• A. Compare the numbers in the last two columns of
  your completed table. Make a conjecture about the
  meaning of x1/2. Use your calculator to test your
  conjecture for several more values of x.
• B. Does your definition work for negative values
  of x ? Why or why not?

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Solution Exploration 5

• A. Compare the numbers in the last two columns of
  your completed table. Make a conjecture about the
  meaning of x1/2. Use your calculator to test your
  conjecture for several more values of x.
• The numbers in the last two columns in step 1 are
  the same. Conjecture
• B. Does your definition work for negative values
  of x ? Why or why not?
• No, only positive numbers have a real square root
  so if x 0, x1/2 is not a real number.

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Exploration 6

• Make a conjecture about the meaning of x1/3. Use
  your calculator to test your conjecture for
  several positive values of x and several negative
  values of x. Does your result support your
  conjecture?

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Solution Exploration 6

• Make a conjecture about the meaning of x1/3. Use
  your calculator to test your conjecture for
  several positive values of x and several negative
  values of x. Does your result support your
  conjecture?
• Conjecture
• Examples

The results support the conjecture
22
Square Roots Cube Roots (expressed as
fractional exponents)
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Talk it Over 1-4

• 1. 811/2
• 2. 271/3
• 3. (-8)1/3
• 4. Let a 9 and b 16. Find the value of ab1/2
  and of (ab)1/2. How do the values compare?

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Solution Talk it Over 1-4

• 1. 811/2
• 2. 271/3
• 3. (-8)1/3
• 4. Let a 9 and b 16. Find the value of ab1/2
  and of (ab)1/2. How do the values compare?
• 9161/2
• (916)1/2

The values are different since in the first
equation a is not raised to the ½ power
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Talk it Over 5-6

• The radical form of the expression (2y)1/3 is
  . Rewrite each expression in the form
  indicated.
• 5. (5x)1/2 in radical form
• 6. in exponential form

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Solution Talk it Over 5-6

• The radical form of the expression (2y)1/3 is
  . Rewrite each expression in the form
  indicated.
• 5. (5x)1/2 in radical form
• 6. in exponential form
• 4n1/3

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Sample 2

• You can use the formula below to estimate the
  speed v (in feet per second) that a roller
  coaster car must travel in order to stay on a
  vertical loop of track with radius r (in feet).
• v (32r)1/2
• About how fast must a roller coaster car travel
  on a vertical loop of track with radius 23 ft?
  Round your answer to the nearest foot per second.

Watch Out! To find the square root of a product,
you first multiply and then find the square root
By the way the Moonsault Scramble coaster at the
Fujikyu Highland Park, near Kawaguchi Lake,
Japan, is 207 ft tall. When it was built, it was
the tallest above-ground roller coaster in the
world.
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Solution Sample 2

• You can use the formula below to estimate the
  speed v (in feet per second) that a roller
  coaster car must travel in order to stay on a
  vertical loop of track with radius r (in feet).
• v (32r)1/2
• About how fast must a roller coaster car travel
  on a vertical loop of track with radius 23 ft?
  Round your answer to the nearest foot per second.
• v (32r)1/2
• (3223)1/2
• (736)1/2
• 27.13

The speed of the roller coaster car must be about
27 ft/s.
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Look Back

• Summarize the meaning of each type of exponent
  positive integer, negative integer, zero, one
  half, and one third.

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Solution Look Back

• Summarize the meaning of each type of exponent
  positive integer, negative integer, zero, one
  half, end one third.
• Positive integer exponents represent powers.
• Negative integer exponents represent reciprocals
  of powers.
• Any number to a zero exponent is 1 except for 00,
  which is undefined.
• An exponent of one-half represents the square
  root.
• An exponent of one-third represents the cube root.

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The End