Which Of The Following Expressions Are Equivalent To $\frac{4 {-3}}{4 {-8}}$?A. $\frac{4 8}{4 3}$B. $\frac{4 {-8}}{4 {-3}}$C. $ 4 − 5 4^{-5} 4 − 5 [/tex]D. $4^5$

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Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will explore the concept of equivalent forms of exponential expressions, focusing on the given expression $\frac{4{-3}}{4{-8}}$. We will examine each option and determine which ones are equivalent to the given expression.

Understanding Exponential Notation

Before we dive into the problem, let's briefly review exponential notation. The expression $a^b$ represents a as raised to the power of b. For example, $2^3$ means 2 multiplied by itself 3 times, which equals 8. When a is a positive number, the exponentiation operation is straightforward. However, when a is a negative number or a fraction, the exponentiation operation can be more complex.

The Given Expression

The given expression is $\frac{4{-3}}{4{-8}}$. To simplify this expression, we need to apply the rules of exponentiation. When dividing two exponential expressions with the same base, we subtract the exponents. Therefore, we can rewrite the given expression as $4^{-3-(-8)}$.

Simplifying the Expression

Now, let's simplify the expression $4^{-3-(-8)}$. When subtracting a negative number, we add the corresponding positive number. Therefore, $-3-(-8) = -3+8 = 5$. So, the expression simplifies to $4^5$.

Evaluating the Options

Now that we have simplified the given expression, let's evaluate each option to determine which ones are equivalent.

Option A: $\frac{48}{43}$

To determine if this option is equivalent, we need to apply the rule of dividing exponential expressions with the same base. When dividing two exponential expressions with the same base, we subtract the exponents. Therefore, we can rewrite the expression as $4^{8-3} = 4^5$. Since this expression is equivalent to the simplified form of the given expression, option A is a correct equivalent form.

Option B: $\frac{4{-8}}{4{-3}}$

To determine if this option is equivalent, we need to apply the rule of dividing exponential expressions with the same base. When dividing two exponential expressions with the same base, we subtract the exponents. Therefore, we can rewrite the expression as $4^{-8-(-3)} = 4^{-8+3} = 4^{-5}$. Since this expression is not equivalent to the simplified form of the given expression, option B is not a correct equivalent form.

Option C: $4^{-5}$

This option is equivalent to the expression we obtained in option B. Therefore, it is not a correct equivalent form of the given expression.

Option D: $4^5$

This option is equivalent to the simplified form of the given expression. Therefore, it is a correct equivalent form.

Conclusion

In conclusion, the correct equivalent forms of the given expression $\frac{4{-3}}{4{-8}}$ are options A and D. Option A is $\frac{48}{43}$, and option D is $4^5$. These expressions are equivalent to the simplified form of the given expression, which is $4^5$. Understanding how to simplify exponential expressions and identify equivalent forms is crucial for solving various mathematical problems. By applying the rules of exponentiation and simplifying expressions, we can determine which options are equivalent to the given expression.

Additional Tips and Examples

  • When dividing two exponential expressions with the same base, subtract the exponents.
  • When simplifying an expression with a negative exponent, rewrite it as a positive exponent by moving the base to the other side of the fraction bar.
  • When simplifying an expression with a fraction as the base, rewrite it as a product of two exponential expressions with the same base.

Practice Problems

  1. Simplify the expression $\frac{24}{26}$.
  2. Simplify the expression $\frac{3{-2}}{3{-5}}$.
  3. Simplify the expression $\frac{53}{5{-2}}$.

Answer Key

  1. 2−22^{-2}

  2. 333^3

  3. 5^5$<br/>

Q: What is the rule for dividing exponential expressions with the same base?

A: When dividing two exponential expressions with the same base, we subtract the exponents. For example, $\frac{am}{an} = a^{m-n}$.

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

A: To simplify an expression with a negative exponent, rewrite it as a positive exponent by moving the base to the other side of the fraction bar. For example, $a^{-m} = \frac{1}{a^m}$.

Q: How do I simplify an expression with a fraction as the base?

A: To simplify an expression with a fraction as the base, rewrite it as a product of two exponential expressions with the same base. For example, $\left(\frac{a}{b}\right)^m = \frac{am}{bm}$.

Q: What is the difference between $a^m$ and $a^{-m}$?

A: $a^m$ represents a as raised to the power of m, while $a^{-m}$ represents the reciprocal of a raised to the power of m. For example, $2^3 = 8$, while $2^{-3} = \frac{1}{8}$.

Q: Can I simplify an expression with a negative exponent by moving the base to the other side of the fraction bar?

A: Yes, you can simplify an expression with a negative exponent by moving the base to the other side of the fraction bar. For example, $\frac{1}{a^m} = a^{-m}$.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, apply the rule of multiplying exponential expressions with the same base. For example, $a^m \cdot a^n = a^{m+n}$.

Q: Can I simplify an expression with multiple exponents by adding or subtracting the exponents?

A: No, you cannot simplify an expression with multiple exponents by adding or subtracting the exponents. Instead, you need to apply the rule of multiplying exponential expressions with the same base.

Q: What is the rule for multiplying exponential expressions with the same base?

A: When multiplying two exponential expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

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

A: To simplify an expression with a zero exponent, we can rewrite it as 1. For example, $a^0 = 1$.

Q: Can I simplify an expression with a zero exponent by moving the base to the other side of the fraction bar?

A: No, you cannot simplify an expression with a zero exponent by moving the base to the other side of the fraction bar. Instead, you can simply rewrite it as 1.

Conclusion

In conclusion, simplifying exponential expressions requires a clear understanding of the rules of exponentiation. By applying the rules of dividing, multiplying, and simplifying exponential expressions, you can simplify complex expressions and solve various mathematical problems. Remember to always follow the order of operations and to simplify expressions with negative exponents by moving the base to the other side of the fraction bar.