Determine if each number is either rational or irrational.
Rational Numbers
Irrational Numbers
DRAG & DROP THE ANSWER
0.78‾0.\overline{78}0.
78
−519-\frac{5}{19}−
19
5
�\piπ
444
969696
20\sqrt{20}
20
100\sqrt{100}
100
Rational Numbers:
- 0.78‾0.\overline{78}0.
- -519-\frac{5}{19}−
19
5
Irrational Numbers:
- ππ
- 444
- 969696
- 20√2020
- 100√100100
Write the following infinite decimal expansion as a fraction. You do not need to simplify.
0.32‾0.\overline{32}0.
32
To express the infinite decimal expansion 0.32‾0.\overline{32}0 as a fraction, we can set up the following equation:
Let x = 0.32‾0.\overline{32}0
Then, multiplying both sides of the equation by 100, we get:
100x = 32.‾32‾32...
Now, let's subtract the original equation from the equation multiplied by 100:
100x - x = 32.‾32‾32... - 0.32‾0.\overline{32}0
99x = 32.‾32‾32... - 0.32‾0.\overline{32}0
To subtract the recurring decimals, we need to align the decimal places. We get:
99x = 32.‾00‾00...
And since 32.‾00 equals 32, we have:
99x = 32
Now, divide both sides of the equation by 99:
x = 32/99
Therefore, the fraction representation of the infinite decimal expansion 0.32‾0.\overline{32}0 is 32/99.
what is 0.323232... simplified
To simplify the infinite decimal expansion 0.323232..., we can set up the following equation:
Let x = 0.323232...
Then, multiplying both sides of the equation by 100, we get:
100x = 32.323232...
Now, let's subtract the original equation from the equation multiplied by 100:
100x - x = 32.323232... - 0.323232...
99x = 32
To find the fraction representation, divide both sides of the equation by 99:
x = 32/99
Therefore, the simplified fraction representation of the infinite decimal expansion 0.323232... is 32/99.
Simplify (3−8)−7\left(3^{-8}\right)^{-7}(3
−8
)
−7
. Write your answer in exponential form. (Example: 525^25
2
not 252525 ). Click Here for Help Video
To simplify the expression (3^(-8))^(-7) in exponential form, we use the rule that states when raising a power to a negative exponent, invert the base and make the exponent positive.
Applying this rule, we have:
(3^(-8))^(-7) = (1/3^8)^(-7) = (1/3^8)^(7*-1) = 1/3^(8*-7) = 1/3^(-56)
Therefore, the expression (3^(-8))^(-7) simplified in exponential form is 1/3^(-56).