IM Alg2.3.3 Lesson: Exponents That Are Unit Fractions
Find a solution to each equation.
[math]x^2=25[/math]
[math]z^2=7[/math]
[math]y^3=8[/math]
[math]w^3=19[/math]
[size=150]Clare said, “I know that [math]9^2=9\cdot9[/math], [math]9^1=9[/math], and [math]9^0=1[/math]. I wonder what [math]9^{\frac{1}{2}}[/math] means?” First, she graphed [math]y=9^x[/math] for some whole number values of [math]x[/math], and estimated [math]9^{\frac{1}{2}}[/math] from the graph.[/size][br][br]Graph the function yourself. What estimate do you get for [math]9^{\frac{1}{2}}[/math]?
Graph the function yourself.
Using the properties of exponents, Clare evaluated [math]9^{\frac{1}{2}}\cdot9^{\frac{1}{2}}[/math]. What did she get?[br]
For that to be true, what must the value of [math]9^{\frac{1}{2}}[/math] be?[br]
[size=150]Diego saw Clare’s work and said, “Now I’m wondering about [math]3^{\frac{1}{2}}[/math].” First he graphed [math]y=3^x[/math] for some whole number values of [math]x[/math], and estimated [math]3^{\frac{1}{2}}[/math] from the graph.[/size][br][br]Graph the function yourself. What estimate do you get for [math]3^{\frac{1}{2}}[/math]?[br]
Graph the function yourself.
Next he used exponent rules to find the value of [math](3^{\frac{1}{2}})^2[/math]. What did he find?[br]
Then he said, “That looks like a root!” What do you think he means?[br]
Use the exponent rules and your understanding of roots to find the exact value of:
[math]25^{\frac{1}{2}}[/math]
[math]15^{\frac{1}{2}}[/math]
[math]8^{\frac{1}{3}}[/math]
[math]2^{\frac{1}{3}}[/math]
Match each exponential expression to an equivalent expression.
How do we know without counting that the number of circles equals the number of squares?
[size=150]Because we can match every circle to exactly one square, and each square has a match:[br][img]data:image/png;base64,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[/img][br]We say that we have shown that there is a one-to-one correspondence of the set of circles and the set of squares. We can do this with infinite sets, too! For example, there are the same “number” of positive integers as there are even positive integers:[br][img]data:image/png;base64,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[/img][br]Every positive integer is matched to exactly one even positive integer, and every even integer has a match! We have shown that there is a one-to-one correspondence between the set of positive integers and the set of even positive integers. Whenever we can make a one-to-one matching like this of the positive integers to another set, we say the other set is [i]countable.[/i][/size][br][br]Show that the set of square roots of positive integers is countable.[br]
Show that the set of positive integer roots of 2 is countable.[br]
Show that the set of positive integer roots of positive integers is countable. (Hint: there is a famous proof that the positive rational numbers are countable. Find and study this proof.)[br]
IM Alg2.3.3 Practice: Exponents That Are Unit Fractions
[size=150]Suppose that a friend missed class and never learned what [math]25^{\frac{1}{2}}[/math] means.[/size][br][br]Use exponent rules your friend would already know to calculate [math]25^{\frac{1}{2}}\cdot25^{\frac{1}{2}}[/math].
Explain why this means that [math]25^{\frac{1}{2}}=5[/math].[br]
[size=150]Which expression is equivalent to [math]16^{\frac{1}{2}}[/math]?[/size]
Select [b]all [/b]the expressions equivalent to [math]4^{10}[/math].
A square has side length [math]\sqrt{82}[/math] cm. What is the area of the square?