Calculate The Following Expression:${ \frac{1.1 \times 10^4 + 42.3 \times 10^4}{1.7 \times 10^{-7}} }$

Mar 11, 2025 by ADMIN 104 views

**Calculating Complex Expressions in Mathematics**

Introduction

In mathematics, complex expressions are a common occurrence in various fields, including algebra, calculus, and physics. These expressions often involve multiple operations, including addition, subtraction, multiplication, and division, and can be represented in various forms, such as scientific notation or exponential notation. In this article, we will focus on calculating a specific complex expression that involves scientific notation and exponential notation.

The Expression to be Calculated

The expression to be calculated is:

{ \frac{1.1 \times 10^4 + 42.3 \times 10^4}{1.7 \times 10^{-7}} \}

This expression involves two terms in the numerator and one term in the denominator. The numerator contains two terms, each of which is a product of a decimal number and a power of 10. The denominator is a product of a decimal number and a negative power of 10.

Step 1: Simplify the Numerator

To simplify the numerator, we need to add the two terms together. However, before we can do that, we need to express both terms in the same form. We can do this by converting the first term to have the same exponent as the second term.

{ 1.1 \times 10^4 = 1.1 \times 10^4 \times 10^0 \}

Now we can add the two terms together:

{ 1.1 \times 10^4 \times 10^0 + 42.3 \times 10^4 \}

Using the properties of exponents, we can combine the two terms into a single term:

{ (1.1 \times 10^4 + 42.3 \times 10^4) \times 10^0 \}

Now we can simplify the expression by combining the two terms:

{ 43.4 \times 10^4 \}

Step 2: Simplify the Denominator

To simplify the denominator, we need to express it in the same form as the numerator. We can do this by converting the term to have the same exponent as the numerator.

{ 1.7 \times 10^{-7} = 1.7 \times 10^{-7} \times 10^0 \}

Now we can simplify the expression by combining the two terms:

{ 1.7 \times 10^{-7} \times 10^0 \}

Step 3: Divide the Numerator by the Denominator

Now that we have simplified the numerator and denominator, we can divide the numerator by the denominator. To do this, we need to use the properties of exponents to simplify the expression.

{ \frac{43.4 \times 10^4}{1.7 \times 10^{-7} \times 10^0} \}

Using the properties of exponents, we can simplify the expression by combining the two terms:

{ \frac{43.4 \times 10^4}{1.7 \times 10^{-7}} \}

Now we can simplify the expression by dividing the numerator by the denominator:

{ \frac{43.4}{1.7} \times 10^{4-(-7)} \}

Using the properties of exponents, we can simplify the expression by combining the two terms:

{ 25.53 \times 10^{11} \}

Conclusion

In this article, we calculated a complex expression that involved scientific notation and exponential notation. We simplified the numerator and denominator by combining the two terms and using the properties of exponents. We then divided the numerator by the denominator to obtain the final result. The final result is a product of a decimal number and a power of 10.

Final Answer

The final answer is:

{ 25.53 \times 10^{11} \}

Discussion

This expression is a common occurrence in various fields, including algebra, calculus, and physics. It involves multiple operations, including addition, subtraction, multiplication, and division, and can be represented in various forms, such as scientific notation or exponential notation. The expression is simplified by combining the two terms in the numerator and using the properties of exponents to simplify the expression.

Real-World Applications

This expression has many real-world applications, including:

Calculating the area of a circle
Calculating the volume of a sphere
Calculating the surface area of a cube
Calculating the volume of a cylinder

These are just a few examples of the many real-world applications of this expression. The expression is a fundamental concept in mathematics and is used in many different fields.

Conclusion

Introduction

In our previous article, we calculated a complex expression that involved scientific notation and exponential notation. We simplified the numerator and denominator by combining the two terms and using the properties of exponents. We then divided the numerator by the denominator to obtain the final result. In this article, we will answer some common questions related to calculating complex expressions in mathematics.

Q: What is the difference between scientific notation and exponential notation?

A: Scientific notation and exponential notation are two different ways of representing numbers in mathematics. Scientific notation is a way of representing numbers in the form a × 10^b, where a is a decimal number between 1 and 10, and b is an integer. Exponential notation is a way of representing numbers in the form a^b, where a is a base number and b is an exponent.

Q: How do I simplify a complex expression with multiple terms in the numerator?

A: To simplify a complex expression with multiple terms in the numerator, you need to combine the terms by adding or subtracting them. You can do this by using the properties of exponents to simplify the expression.

Q: What is the rule for dividing numbers with different exponents?

A: When dividing numbers with different exponents, you need to subtract the exponents. For example, if you have the expression 10^a / 10^b, you can simplify it by subtracting the exponents: 10^(a-b).

Q: How do I handle negative exponents?

A: When you have a negative exponent, you can handle it by taking the reciprocal of the base number. For example, if you have the expression 10^(-a), you can simplify it by taking the reciprocal of the base number: 1/10^a.

Q: What is the difference between a decimal number and a fraction?

A: A decimal number is a number that is represented in the form a.b, where a is the whole number part and b is the fractional part. A fraction is a number that is represented in the form a/b, where a is the numerator and b is the denominator.

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

A: To convert a decimal number to a fraction, you need to express the decimal number as a ratio of two integers. You can do this by using the properties of fractions to simplify the expression.

Q: What is the rule for multiplying numbers with different exponents?

A: When multiplying numbers with different exponents, you need to add the exponents. For example, if you have the expression 10^a × 10^b, you can simplify it by adding the exponents: 10^(a+b).

Q: How do I handle complex expressions with multiple operations?

A: When you have a complex expression with multiple operations, you need to follow the order of operations (PEMDAS) to simplify the expression. This means that you need to perform the operations in the following order: parentheses, exponents, multiplication and division, and addition and subtraction.

Conclusion

In conclusion, calculating complex expressions in mathematics is a fundamental concept that is used in many different fields. We have answered some common questions related to calculating complex expressions in mathematics, including the difference between scientific notation and exponential notation, how to simplify a complex expression with multiple terms in the numerator, and how to handle negative exponents. We hope that this article has been helpful in answering your questions and providing you with a better understanding of calculating complex expressions in mathematics.

Final Answer

The final answer is:

{ 25.53 \times 10^{11} \}

Discussion

Real-World Applications

This expression has many real-world applications, including:

Calculating the area of a circle
Calculating the volume of a sphere
Calculating the surface area of a cube
Calculating the volume of a cylinder

These are just a few examples of the many real-world applications of this expression. The expression is a fundamental concept in mathematics and is used in many different fields.