Rewrite Each Of The Following Using A Single Exponential Expression.(b) { \frac 3 {5x}}{3 {x-2}}$}$Options 1. ${$3^{5x(x-2) $}$2. ${$3^{5x-x-2}$}$3. ${ 3\$} (c)
Introduction
Exponential expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on rewriting exponential expressions using a single exponential expression. We will explore the concept of exponential expressions, the rules for simplifying them, and provide step-by-step examples to illustrate the process.
Understanding Exponential Expressions
Exponential expressions are mathematical expressions that involve a base raised to a power. The base is the number that is being raised to the power, and the exponent is the number that is being raised to. For example, in the expression 3^5, 3 is the base and 5 is the exponent.
The Quotient Rule for Exponents
One of the most important rules for simplifying exponential expressions is the quotient rule. The quotient rule states that when we divide two exponential expressions with the same base, we can subtract the exponents. In other words, a^(m-n) = a^m / a^n.
Rewriting the Given Expression
Now, let's apply the quotient rule to rewrite the given expression:
{\frac{3{5x}}{3{x-2}}$}$
Using the quotient rule, we can rewrite the expression as:
${3^{5x-(x-2)}\$}
Simplifying the Expression
Now, let's simplify the expression by combining the exponents:
${3^{5x-(x-2)}\$}
= ${3^{5x-x+2}\$}
= ${3^{4x+2}\$}
Therefore, the rewritten expression is:
${3^{4x+2}\$}
Conclusion
In this article, we have learned how to rewrite exponential expressions using a single exponential expression. We have applied the quotient rule to simplify the given expression and have arrived at the final answer. By following the steps outlined in this article, you can simplify any exponential expression and rewrite it in a single exponential form.
Common Mistakes to Avoid
When rewriting exponential expressions, it's essential to avoid common mistakes. Here are a few mistakes to watch out for:
- Not applying the quotient rule: Make sure to apply the quotient rule when dividing two exponential expressions with the same base.
- Not simplifying the expression: Don't forget to simplify the expression by combining the exponents.
- Not checking the final answer: Always check the final answer to ensure that it's correct.
Practice Problems
To practice rewriting exponential expressions, try the following problems:
- Rewrite the expression {\frac{2{3x}}{2{x-1}}$}$ using a single exponential expression.
- Rewrite the expression {\frac{5{2x}}{5{x-3}}$}$ using a single exponential expression.
- Rewrite the expression {\frac{3{4x}}{3{x-2}}$}$ using a single exponential expression.
Answer Key
- ${2^{3x-(x-1)}\$} = ${2^{2x+1}\$}
- ${5^{2x-(x-3)}\$} = ${5^{x+3}\$}
- ${3^{4x-(x-2)}\$} = ${3^{3x+2}\$}
Q: What is the quotient rule for exponents?
A: The quotient rule for exponents states that when we divide two exponential expressions with the same base, we can subtract the exponents. In other words, a^(m-n) = a^m / a^n.
Q: How do I apply the quotient rule to rewrite an exponential expression?
A: To apply the quotient rule, simply subtract the exponents of the two expressions. For example, if we have the expression {\frac{3{5x}}{3{x-2}}$}$, we can rewrite it as ${3^{5x-(x-2)}\$}.
Q: What is the difference between a base and an exponent?
A: The base is the number that is being raised to the power, and the exponent is the number that is being raised to. For example, in the expression 3^5, 3 is the base and 5 is the exponent.
Q: Can I rewrite an exponential expression with a different base?
A: Yes, you can rewrite an exponential expression with a different base. However, you will need to use the change of base formula, which is a^x = (bx)(log_a(b)).
Q: How do I simplify an exponential expression?
A: To simplify an exponential expression, you can combine the exponents by adding or subtracting them. For example, if we have the expression ${3^{4x+2}\$}, we can simplify it by combining the exponents.
Q: What are some common mistakes to avoid when rewriting exponential expressions?
A: Some common mistakes to avoid when rewriting exponential expressions include:
- Not applying the quotient rule
- Not simplifying the expression
- Not checking the final answer
Q: How can I practice rewriting exponential expressions?
A: You can practice rewriting exponential expressions by trying the following problems:
- Rewrite the expression {\frac{2{3x}}{2{x-1}}$}$ using a single exponential expression.
- Rewrite the expression {\frac{5{2x}}{5{x-3}}$}$ using a single exponential expression.
- Rewrite the expression {\frac{3{4x}}{3{x-2}}$}$ using a single exponential expression.
Q: What are some real-world applications of rewriting exponential expressions?
A: Rewriting exponential expressions has many real-world applications, including:
- Calculating population growth
- Modeling financial investments
- Analyzing scientific data
Q: Can I use a calculator to rewrite exponential expressions?
A: Yes, you can use a calculator to rewrite exponential expressions. However, it's essential to understand the underlying math and be able to apply the rules for rewriting exponential expressions.
Q: How can I check my work when rewriting exponential expressions?
A: To check your work, you can plug in values for the variables and see if the expression simplifies correctly. You can also use a calculator to check your work.
Q: What are some advanced topics related to rewriting exponential expressions?
A: Some advanced topics related to rewriting exponential expressions include:
- Exponential functions
- Logarithmic functions
- Exponential equations
By understanding the rules for rewriting exponential expressions and practicing with examples, you can become proficient in this important math concept.