Calculate The Following Expression:b) 9 × 10 7 3 × 10 3 \frac{9 \times 10^7}{3 \times 10^3} 3 × 1 0 3 9 × 1 0 7 ​

Introduction

In mathematics, exponential expressions are a fundamental concept that helps us represent very large or very small numbers in a compact form. When dealing with exponential expressions, it's essential to understand the rules of exponents and how to simplify them. In this article, we will focus on simplifying exponential expressions, specifically the expression $\frac{9 \times 10^7}{3 \times 10^3}$.

Understanding Exponential Notation

Exponential notation is a shorthand way of representing numbers with a base and an exponent. The base is the number being raised to a power, and the exponent is the power to which the base is raised. For example, $10^7$ represents $10$ raised to the power of $7$, which is equal to $10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10,000,000$.

Simplifying Exponential Expressions

When simplifying exponential expressions, we need to follow the rules of exponents. The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, $10^7 \times 10^3 = 10^{7+3} = 10^{10}$.

**Simplifying the Expression $\frac{9 \times 10^7}{3 \times 10^3}$

To simplify the expression $\frac{9 \times 10^7}{3 \times 10^3}$, we need to follow the rules of exponents. We can start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator is $9 \times 10^7$. We can simplify this by dividing $9$ by $3$, which gives us $3$. So, the numerator becomes $3 \times 10^7$.

Simplifying the Denominator

The denominator is $3 \times 10^3$. We can simplify this by dividing $3$ by $3$, which gives us $1$. So, the denominator becomes $1 \times 10^3$.

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator. This gives us:

$$\frac{3 \times 10^7}{1 \times 10^3} = 3 \times 10^{7-3} = 3 \times 10^4$$

Evaluating the Expression

To evaluate the expression $3 \times 10^4$, we need to multiply $3$ by $10^4$. This gives us:

$$3 \times 10^4 = 3 \times 100,000 = 300,000$$

Conclusion

In this article, we simplified the exponential expression $\frac{9 \times 10^7}{3 \times 10^3}$ by following the rules of exponents. We simplified the numerator and denominator separately and then divided the numerator by the denominator to get the final result. The final result is $300,000$.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. These include:

• Not following the rules of exponents: When simplifying exponential expressions, it's essential to follow the rules of exponents. This includes adding exponents when multiplying numbers with the same base and subtracting exponents when dividing numbers with the same base.
• Not simplifying the numerator and denominator separately: When simplifying exponential expressions, it's essential to simplify the numerator and denominator separately before dividing them.
• Not evaluating the expression correctly: When evaluating an exponential expression, it's essential to multiply the base by the exponent to get the final result.

Real-World Applications

Exponential expressions have numerous real-world applications. These include:

• Finance: Exponential expressions are used in finance to calculate interest rates and investment returns.
• Science: Exponential expressions are used in science to calculate the growth and decay of populations and the spread of diseases.
• Engineering: Exponential expressions are used in engineering to calculate the stress and strain on materials and the efficiency of systems.

Final Thoughts

Introduction

In our previous article, we discussed how to simplify exponential expressions by following the rules of exponents. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying exponential expressions.

Q: What are the rules of exponents?

A: The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, $10^7 \times 10^3 = 10^{7+3} = 10^{10}$. When we divide two numbers with the same base, we subtract their exponents. For example, $\frac{10^7}{10^3} = 10^{7-3} = 10^4$.

Q: How do I simplify an exponential expression with a negative exponent?

A: To simplify an exponential expression with a negative exponent, we can rewrite the expression with a positive exponent by moving the base to the other side of the fraction. For example, $10^{-3} = \frac{1}{10^3}$.

Q: Can I simplify an exponential expression with a variable exponent?

A: Yes, you can simplify an exponential expression with a variable exponent by following the rules of exponents. For example, $x^{a+b} = x^a \times x^b$.

Q: How do I simplify an exponential expression with a fraction exponent?

A: To simplify an exponential expression with a fraction exponent, we can rewrite the expression with a positive exponent by multiplying the base by the reciprocal of the fraction. For example, $10^{\frac{1}{2}} = \sqrt{10}$.

Q: Can I simplify an exponential expression with a decimal exponent?

A: Yes, you can simplify an exponential expression with a decimal exponent by following the rules of exponents. For example, $10^{3.5} = 10^3 \times 10^{0.5} = 1000 \times \sqrt{10}$.

Q: How do I simplify an exponential expression with a negative base?

A: To simplify an exponential expression with a negative base, we can rewrite the expression with a positive base by changing the sign of the exponent. For example, $(-10)^3 = -10^3$.

Q: Can I simplify an exponential expression with a complex number base?

A: Yes, you can simplify an exponential expression with a complex number base by following the rules of exponents. For example, $(2+3i)^4 = (2+3i)^2 \times (2+3i)^2 = (4+12i+9i^2)^2 = (-5+12i)^2$.

Q: How do I simplify an exponential expression with a variable base?

A: To simplify an exponential expression with a variable base, we can rewrite the expression with a positive base by changing the sign of the exponent. For example, $x^{-3} = \frac{1}{x^3}$.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying exponential expressions. By following the rules of exponents and simplifying the numerator and denominator separately, you can simplify complex exponential expressions and get the final result. Remember to always follow the rules of exponents and to simplify the numerator and denominator separately to get the correct result.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. These include:

• Not following the rules of exponents: When simplifying exponential expressions, it's essential to follow the rules of exponents. This includes adding exponents when multiplying numbers with the same base and subtracting exponents when dividing numbers with the same base.
• Not simplifying the numerator and denominator separately: When simplifying exponential expressions, it's essential to simplify the numerator and denominator separately before dividing them.
• Not evaluating the expression correctly: When evaluating an exponential expression, it's essential to multiply the base by the exponent to get the final result.

Real-World Applications

Exponential expressions have numerous real-world applications. These include:

• Finance: Exponential expressions are used in finance to calculate interest rates and investment returns.
• Science: Exponential expressions are used in science to calculate the growth and decay of populations and the spread of diseases.
• Engineering: Exponential expressions are used in engineering to calculate the stress and strain on materials and the efficiency of systems.

Final Thoughts

In conclusion, simplifying exponential expressions is a crucial skill in mathematics. By following the rules of exponents and simplifying the numerator and denominator separately, you can simplify complex exponential expressions and get the final result. Exponential expressions have numerous real-world applications, and understanding how to simplify them is essential for success in finance, science, and engineering.