Simplify { \frac{4 8}{4 3}$}$.Express Using Positive Exponents.A. ${ 4^{11}\$} B. ${ 4^{24}\$} C. ${ 4^5\$} D. ${ 4^{-5}\$}
Simplify {\frac{48}{43}$}$ and Express Using Positive Exponents
Understanding Exponents and Simplifying Expressions
Exponents are a fundamental concept in mathematics, and they play a crucial role in simplifying complex expressions. In this article, we will focus on simplifying the expression {\frac{48}{43}$}$ and express it using positive exponents.
What are Exponents?
Exponents are a shorthand way of representing repeated multiplication. For example, ${4^3\$} can be read as "4 to the power of 3" and is equivalent to ${4 \times 4 \times 4\$}. Exponents can be positive or negative, and they can also be fractions.
Simplifying the Expression {\frac{48}{43}$}$
To simplify the expression {\frac{48}{43}$}$, we need to apply the quotient rule of exponents. The quotient rule states that when we divide two powers with the same base, we subtract the exponents.
Quotient Rule of Exponents
The quotient rule of exponents states that:
{\frac{am}{an} = a^{m-n}$}$
where {a$}$ is the base and {m$}$ and {n$}$ are the exponents.
Applying the Quotient Rule
In our expression, the base is ${4\$} and the exponents are ${8\$} and ${3\$}. Applying the quotient rule, we get:
{\frac{48}{43} = 4^{8-3} = 4^5$}$
Expressing Using Positive Exponents
The expression ${4^5\$} is already in positive exponent form. However, we can also express it in a different way by using the product rule of exponents.
Product Rule of Exponents
The product rule of exponents states that when we multiply two powers with the same base, we add the exponents.
Expressing Using the Product Rule
We can express ${4^5\$} as ${4^3 \times 4^2\$}. Applying the product rule, we get:
${4^5 = 4^3 \times 4^2 = 4^{3+2} = 4^5\$}
Conclusion
In this article, we simplified the expression {\frac{48}{43}$}$ using the quotient rule of exponents and expressed it using positive exponents. We also explored the product rule of exponents and how it can be used to express the same expression in a different way.
Answer Options
Based on our simplification, the correct answer is:
- C. ${4^5\$}
The other options are incorrect because:
- A. ${4^{11}\$} is incorrect because the exponent is not ${11\$}.
- B. ${4^{24}\$} is incorrect because the exponent is not ${24\$}.
- D. ${4^{-5}\$} is incorrect because the exponent is negative.
Final Answer
The final answer is C. ${4^5\$}.
Simplify {\frac{48}{43}$}$ and Express Using Positive Exponents: Q&A
Understanding Exponents and Simplifying Expressions
Exponents are a fundamental concept in mathematics, and they play a crucial role in simplifying complex expressions. In this article, we will focus on simplifying the expression {\frac{48}{43}$}$ and express it using positive exponents.
Q&A: Simplifying Exponents
Q: What is the quotient rule of exponents?
A: The quotient rule of exponents states that when we divide two powers with the same base, we subtract the exponents. This can be represented as:
{\frac{am}{an} = a^{m-n}$}$
Q: How do I apply the quotient rule to simplify the expression {\frac{48}{43}$}$?
A: To simplify the expression {\frac{48}{43}$}$, we need to apply the quotient rule by subtracting the exponents. This gives us:
{\frac{48}{43} = 4^{8-3} = 4^5$}$
Q: What is the product rule of exponents?
A: The product rule of exponents states that when we multiply two powers with the same base, we add the exponents. This can be represented as:
{a^m \times a^n = a^{m+n}$}$
Q: How do I express ${4^5\$} using the product rule?
A: We can express ${4^5\$} as ${4^3 \times 4^2\$}. Applying the product rule, we get:
${4^5 = 4^3 \times 4^2 = 4^{3+2} = 4^5\$}
Q: What is the difference between positive and negative exponents?
A: Positive exponents represent a power that is raised to a positive number, while negative exponents represent a power that is raised to a negative number. For example, ${4^3\$} is a positive exponent, while ${4^{-3}\$} is a negative exponent.
Q: How do I simplify expressions with negative exponents?
A: To simplify expressions with negative exponents, we can use the rule that states:
{a^{-n} = \frac{1}{a^n}$}$
This means that a negative exponent can be rewritten as a fraction with the base in the denominator.
Q: Can I use the quotient rule to simplify expressions with negative exponents?
A: Yes, you can use the quotient rule to simplify expressions with negative exponents. For example, if we have the expression {\frac{4{-3}}{4{-2}}$}$, we can apply the quotient rule by subtracting the exponents:
{\frac{4{-3}}{4{-2}} = 4^{-3-(-2)} = 4^{-1}$}$
Conclusion
In this article, we simplified the expression {\frac{48}{43}$}$ using the quotient rule of exponents and expressed it using positive exponents. We also explored the product rule of exponents and how it can be used to express the same expression in a different way. Additionally, we discussed the difference between positive and negative exponents and how to simplify expressions with negative exponents.
Answer Options
Based on our simplification, the correct answer is:
- C. ${4^5\$}
The other options are incorrect because:
- A. ${4^{11}\$} is incorrect because the exponent is not ${11\$}.
- B. ${4^{24}\$} is incorrect because the exponent is not ${24\$}.
- D. ${4^{-5}\$} is incorrect because the exponent is negative.
Final Answer
The final answer is C. ${4^5\$}.