Determine each equation is it rational or irrational
The square root of 25
0.666...
Square root of 11
4/7
Let x be equal to the decimal 0.5555...
We can convert this repeating decimal to a fraction by multiplying it by a power of 10 to eliminate the repeating part.
10x = 5.5555...
Now, to get rid of the repeating part, we'll subtract the original equation from this new one:
10x - x = 5.5555... - 0.5555...
Simplifying:
9x = 5
Dividing both sides by 9 to solve for x:
x = 5/9
Therefore, the fraction form of 0.5555... is 5/9.
The square root of 25 is rational because 25 is a perfect square.
0.666... is rational because it can be expressed as a fraction (2/3).
The square root of 11 is irrational because 11 is not a perfect square.
4/7 is rational because it can be expressed as a fraction.
What is 121212... as a fully simplified fraction
To determine the fraction form of the repeating decimal 0.121212..., we will let x be equal to the decimal. Hence, we can express it as:
x = 0.121212...
Multiplying both sides by 100 to “shift” the decimal point:
100x = 12.121212...
Subtracting the original equation from this new one:
100x - x = 12.121212... - 0.121212...
Simplifying:
99x = 12
Dividing by 99 on both sides to solve for x:
x = 12/99
This fraction can be further simplified by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 3:
x = (12/3)/(99/3)
x = 4/33
Therefore, the fully simplified fraction form of 0.121212... is 4/33.
What is 0.5555... as a fraction
Evaluate 8×8-6×2
To evaluate the expression 8×8-6×2, we can use the order of operations.
First, we need to perform the multiplications:
8 × 8 = 64
6 × 2 = 12
Then, we subtract the result of the second multiplication from the first multiplication:
64 - 12 = 52
Therefore, 8×8-6×2 equals 52.
Evaluate -4/2^2-(7-4)
To evaluate the expression -4/2^2-(7-4), we follow the order of operations (PEMDAS/BODMAS).
First, we simplify the exponent:
2^2 = 2 × 2 = 4.
Next, we simplify the inner parentheses:
7 - 4 = 3.
Now, we substitute the simplified values back into the expression:
-4/4 - 3.
Finally, we perform the division and subtraction:
-1 - 3 = -4.
Therefore, -4/2^2 - (7-4) equals -4.
To determine whether each equation represents a rational or irrational number, we need to understand the definitions of these terms.
A rational number is any number that can be expressed as a ratio of two integers (where the denominator is not zero). This means that a rational number can be written as a fraction.
An irrational number, on the other hand, cannot be expressed as the ratio of two integers. Irrational numbers are non-repeating and non-terminating decimals.
Now, let's analyze each equation:
1. The square root of 25:
The square root of 25 is 5. Since the number 5 can be expressed as the fraction 5/1, it is a rational number. Therefore, the equation "The square root of 25" represents a rational number.
2. 0.666... (repeating decimal):
0.666... is a decimal representation of the fraction 2/3. You can convert it to a fraction by assuming x = 0.666..., then multiplying both sides by 10 to get 10x = 6.666..., and subtracting the original equation from the new one, 10x - x = 6.666... - 0.666..., simplifying to 9x = 6. The final step is to solve for x, so x = 6/9 which reduces to 2/3. Since 2/3 is a fraction, 0.666... is a rational number.
3. The square root of 11:
The square root of 11 is a non-repeating and non-terminating decimal. It cannot be expressed as the ratio of two integers. Hence, the square root of 11 is an irrational number. Therefore, the equation "The square root of 11" represents an irrational number.
4. 4/7:
4/7 is a fraction, and any fraction can be expressed as the ratio of two integers. Therefore, 4/7 is a rational number. Hence, the equation "4/7" represents a rational number.
To summarize:
- "The square root of 25" represents a rational number.
- "0.666..." represents a rational number.
- "The square root of 11" represents an irrational number.
- "4/7" represents a rational number.