Practice: Simplify The Expression And Write Your Answer As A Power.1. { \frac{7.6 {13}}{7.6 3}$}$2. { \frac{u {33}}{u {11}}$}$3. One Kilometer Equals ${ 10^3\$} Meters. One Tetrameter Equals ${ 10^{12}\$} Meters.
Introduction
Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore three different exponential expressions and provide step-by-step solutions to simplify them. We will also discuss the importance of understanding exponential expressions and how they are used in real-world applications.
Simplifying Exponential Expressions
1. Simplifying the Expression {\frac{7.6{13}}{7.63}$}$
To simplify the expression {\frac{7.6{13}}{7.63}$}$, we can use the quotient rule of exponents, which states that when we divide two exponential expressions with the same base, we can subtract the exponents.
Step 1: Identify the base and the exponents
The base of the expression is 7.6, and the exponents are 13 and 3.
Step 2: Apply the quotient rule of exponents
Using the quotient rule, we can rewrite the expression as:
{\frac{7.6{13}}{7.63} = 7.6^{13-3} = 7.6^{10}$}$
Therefore, the simplified expression is ${7.6^{10}\$}.
2. Simplifying the Expression {\frac{u{33}}{u{11}}$}$
To simplify the expression {\frac{u{33}}{u{11}}$}$, we can use the quotient rule of exponents, which states that when we divide two exponential expressions with the same base, we can subtract the exponents.
Step 1: Identify the base and the exponents
The base of the expression is u, and the exponents are 33 and 11.
Step 2: Apply the quotient rule of exponents
Using the quotient rule, we can rewrite the expression as:
{\frac{u{33}}{u{11}} = u^{33-11} = u^{22}$}$
Therefore, the simplified expression is {u^{22}$}$.
3. Simplifying the Expression {\frac{10{12}}{103}$}$
To simplify the expression {\frac{10{12}}{103}$}$, we can use the quotient rule of exponents, which states that when we divide two exponential expressions with the same base, we can subtract the exponents.
Step 1: Identify the base and the exponents
The base of the expression is 10, and the exponents are 12 and 3.
Step 2: Apply the quotient rule of exponents
Using the quotient rule, we can rewrite the expression as:
{\frac{10{12}}{103} = 10^{12-3} = 10^9$}$
Therefore, the simplified expression is ${10^9\$}.
Real-World Applications of Exponential Expressions
Exponential expressions are used in a wide range of real-world applications, including:
- Finance: Exponential expressions are used to calculate compound interest and investment returns.
- Science: Exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Exponential expressions are used to design and optimize systems, such as electronic circuits and mechanical systems.
Conclusion
In conclusion, simplifying exponential expressions is a crucial skill to master in mathematics. By understanding the quotient rule of exponents and applying it to different expressions, we can simplify complex expressions and gain a deeper understanding of exponential functions. We hope that this article has provided a comprehensive guide to simplifying exponential expressions and has inspired readers to explore the many real-world applications of exponential expressions.
Additional Resources
For further learning, we recommend the following resources:
- Math textbooks: "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by James Stewart.
- Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
- Practice problems: Try simplifying the following expressions: {\frac{25}{22}$}$, {\frac{37}{34}$}$, and {\frac{4{11}}{45}$}$.
Final Thoughts
Q: What is the quotient rule of exponents?
A: The quotient rule of exponents states that when we divide two exponential expressions with the same base, we can subtract the exponents. This means that {\frac{am}{an} = a^{m-n}$}$.
Q: How do I simplify an exponential expression with a negative exponent?
A: To simplify an exponential expression with a negative exponent, we can rewrite the expression as a fraction with a positive exponent. For example, {a^{-n} = \frac{1}{a^n}$}$.
Q: Can I simplify an exponential expression with a variable exponent?
A: Yes, you can simplify an exponential expression with a variable exponent by applying the quotient rule of exponents. For example, {\frac{am}{an} = a^{m-n}$}$.
Q: How do I simplify an exponential expression with a coefficient?
A: To simplify an exponential expression with a coefficient, we can multiply the coefficient by the exponential expression. For example, ${2a^m = 2a^m\$}.
Q: Can I simplify an exponential expression with a radical?
A: Yes, you can simplify an exponential expression with a radical by applying the quotient rule of exponents. For example, {\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$}$.
Q: How do I simplify an exponential expression with a fraction exponent?
A: To simplify an exponential expression with a fraction exponent, we can apply the quotient rule of exponents. For example, {a^{\frac{m}{n}} = \sqrt[n]{a^m}$}$.
Q: Can I simplify an exponential expression with a negative base?
A: Yes, you can simplify an exponential expression with a negative base by applying the quotient rule of exponents. For example, {(-a)^m = a^m$}$.
Q: How do I simplify an exponential expression with a zero exponent?
A: To simplify an exponential expression with a zero exponent, we can apply the rule that {a^0 = 1$}$.
Q: Can I simplify an exponential expression with a variable base and exponent?
A: Yes, you can simplify an exponential expression with a variable base and exponent by applying the quotient rule of exponents. For example, {\frac{am}{an} = a^{m-n}$}$.
Q: How do I simplify an exponential expression with a mixed exponent?
A: To simplify an exponential expression with a mixed exponent, we can apply the quotient rule of exponents. For example, {a^{m+n} = a^m \cdot a^n$}$.
Conclusion
In conclusion, simplifying exponential expressions is a crucial skill in mathematics that has many real-world applications. By understanding the quotient rule of exponents and applying it to different expressions, we can simplify complex expressions and gain a deeper understanding of exponential functions. We hope that this article has provided a comprehensive guide to simplifying exponential expressions and has inspired readers to explore the many real-world applications of exponential expressions.
Additional Resources
For further learning, we recommend the following resources:
- Math textbooks: "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by James Stewart.
- Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
- Practice problems: Try simplifying the following expressions: {\frac{25}{22}$}$, {\frac{37}{34}$}$, and {\frac{4{11}}{45}$}$.
Final Thoughts
Simplifying exponential expressions is a fundamental skill in mathematics that has many real-world applications. By mastering this skill, we can gain a deeper understanding of exponential functions and apply them to a wide range of problems. We hope that this article has provided a comprehensive guide to simplifying exponential expressions and has inspired readers to explore the many real-world applications of exponential expressions.