1. Between Which Two Consecutive Integers Does $\sqrt{138}$ Lie?2. Rewrite $0.2\dot{6}$ As A Proper Fraction (in The Form Of $\frac{a}{b}$), And Show All Steps.3. Rewrite 92 And 146 As Products Of Their Prime Factors.4. Write

Problem 1: Finding the Consecutive Integers for $\sqrt{138}$

Introduction

In this problem, we are tasked with finding the two consecutive integers between which the square root of 138 lies. To solve this, we will first find the square root of 138 and then determine the two consecutive integers that it falls between.

Step 1: Finding the Square Root of 138

The square root of 138 can be found using a calculator or by using the long division method. Using a calculator, we find that $\sqrt{138} \approx 11.73$.

Step 2: Determining the Consecutive Integers

Now that we have the approximate value of the square root of 138, we can determine the two consecutive integers that it falls between. Since $\sqrt{138} \approx 11.73$, it lies between the integers 11 and 12.

Conclusion

Therefore, the two consecutive integers between which $\sqrt{138}$ lies are 11 and 12.

Problem 2: Rewriting $0.2\dot{6}$ as a Proper Fraction

Introduction

In this problem, we are tasked with rewriting the repeating decimal $0.2\dot{6}$ as a proper fraction in the form of $\frac{a}{b}$. To solve this, we will first identify the repeating pattern in the decimal and then use algebraic manipulation to convert it into a fraction.

Step 1: Identifying the Repeating Pattern

The repeating pattern in the decimal $0.2\dot{6}$ is the digit 6. This means that the decimal can be written as $0.226666...$.

Step 2: Setting Up an Equation

Let $x = 0.2\dot{6}$. Multiplying both sides of the equation by 10, we get $10x = 2.2\dot{6}$. Subtracting the original equation from this new equation, we get $9x = 2.0$, which simplifies to $x = \frac{2}{9}$.

Conclusion

Therefore, the repeating decimal $0.2\dot{6}$ can be rewritten as the proper fraction $\frac{2}{9}$.

Problem 3: Rewriting 92 and 146 as Products of Their Prime Factors

Introduction

In this problem, we are tasked with rewriting the numbers 92 and 146 as products of their prime factors. To solve this, we will first find the prime factorization of each number.

Step 1: Finding the Prime Factorization of 92

The prime factorization of 92 can be found by dividing it by the smallest prime number, which is 2. We find that $92 = 2 \times 46$. Continuing to divide by 2, we get $46 = 2 \times 23$. Therefore, the prime factorization of 92 is $2^2 \times 23$.

Step 2: Finding the Prime Factorization of 146

The prime factorization of 146 can be found by dividing it by the smallest prime number, which is 2. We find that $146 = 2 \times 73$. Therefore, the prime factorization of 146 is $2 \times 73$.

Conclusion

Therefore, the number 92 can be rewritten as the product of its prime factors as $2^2 \times 23$, and the number 146 can be rewritten as the product of its prime factors as $2 \times 73$.

Problem 4: Writing a Number in Scientific Notation

Introduction

In this problem, we are tasked with writing the number 0.000456 in scientific notation. To solve this, we will first move the decimal point to the right until we have a number between 1 and 10.

Step 1: Moving the Decimal Point

Moving the decimal point 6 places to the right, we get 456.

Step 2: Writing in Scientific Notation

Since we moved the decimal point 6 places to the right, we can write the number in scientific notation as $4.56 \times 10^{-4}$.

Conclusion

Therefore, the number 0.000456 can be written in scientific notation as $4.56 \times 10^{-4}$.

Conclusion

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Frequently Asked Questions

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., it can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is non-zero. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers, i.e., it cannot be written in the form $\frac{a}{b}$.

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, you can use the following steps:

1. Identify the repeating pattern in the decimal.
2. Set up an equation using the repeating pattern.
3. Solve the equation to find the fraction.

For example, to convert the decimal $0.2\dot{6}$ to a fraction, you can set up the equation $x = 0.2\dot{6}$ and then multiply both sides by 10 to get $10x = 2.2\dot{6}$. Subtracting the original equation from this new equation, you get $9x = 2.0$, which simplifies to $x = \frac{2}{9}$.

Q: What is the difference between a prime number and a composite number?

A: A prime number is a positive integer that is divisible only by itself and 1. A composite number, on the other hand, is a positive integer that is divisible by at least one other number besides itself and 1.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you can use the following steps:

1. Divide the number by the smallest prime number, which is 2.
2. Continue to divide the number by 2 until you can no longer do so.
3. Move on to the next prime number, which is 3, and repeat the process.
4. Continue this process until you have found all the prime factors of the number.

For example, to find the prime factorization of the number 92, you can start by dividing it by 2 to get $92 = 2 \times 46$. Continuing to divide by 2, you get $46 = 2 \times 23$. Therefore, the prime factorization of 92 is $2^2 \times 23$.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed on the number line, i.e., it is a number that can be written in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. An imaginary number, on the other hand, is a number that cannot be expressed on the number line, i.e., it is a number that can be written in the form $bi$, where $b$ is a real number and $i$ is the imaginary unit.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you can use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator.
2. Simplify the resulting expression.

For example, to simplify the complex fraction $\frac{1 + i}{1 - i}$, you can multiply the numerator and denominator by the conjugate of the denominator, which is $1 + i$. This gives you $\frac{(1 + i)(1 + i)}{(1 - i)(1 + i)} = \frac{1 + 2i + i^2}{1 - i^2} = \frac{1 + 2i - 1}{1 + 1} = \frac{2i}{2} = i$.

Conclusion

In this article, we have answered some frequently asked questions in mathematics. We have discussed the difference between rational and irrational numbers, how to convert decimals to fractions, the difference between prime and composite numbers, how to find the prime factorization of a number, the difference between real and imaginary numbers, and how to simplify complex fractions.