Which Is The Value Of This Expression When A = 5 A=5 A = 5 And K = − 2 K=-2 K = − 2 ? ( 3 2 A − 2 3 A − 1 ) K \left(\frac{3^2 A^{-2}}{3 A^{-1}}\right)^k ( 3 A − 1 3 2 A − 2 ​ ) K A. 1 75 \frac{1}{75} 75 1 ​ B. 9 25 \frac{9}{25} 25 9 ​ C. 25 9 \frac{25}{9} 9 25 ​ D. 75

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various fields, including science, engineering, and economics. In this article, we will delve into the world of exponential expressions and explore how to solve them using real-world examples.

What are Exponential Expressions?

Exponential expressions are mathematical expressions that involve the use of exponents, which are shorthand notations for repeated multiplication. Exponents are used to represent the power to which a base number is raised. For example, the expression $3^2$ represents $3$ multiplied by itself $2$ times, or $3 \times 3 = 9$.

The Given Expression

The given expression is $\left(\frac{3^2 a^{-2}}{3 a^{-1}}\right)^k$. To solve this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Simplify the resulting expression.
3. Raise the simplified expression to the power of $k$.

Step 1: Evaluate the Expressions Inside the Parentheses

To evaluate the expressions inside the parentheses, we need to follow the order of operations (PEMDAS). We will start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator is $3^2 a^{-2}$. Using the property of exponents that states $a^m \times a^n = a^{m+n}$, we can simplify the numerator as follows:

$$3^2 a^{-2} = 3^2 \times a^{-2} = 9 \times \frac{1}{a^2} = \frac{9}{a^2}$$

Simplifying the Denominator

The denominator is $3 a^{-1}$. Using the property of exponents that states $a^m \div a^n = a^{m-n}$, we can simplify the denominator as follows:

$$3 a^{-1} = 3 \times \frac{1}{a} = \frac{3}{a}$$

Simplifying the Expression Inside the Parentheses

Now that we have simplified the numerator and denominator, we can simplify the expression inside the parentheses as follows:

$$\frac{3^2 a^{-2}}{3 a^{-1}} = \frac{\frac{9}{a^2}}{\frac{3}{a}} = \frac{9}{a^2} \times \frac{a}{3} = \frac{9a}{3a^2} = \frac{3}{a}$$

Step 2: Simplify the Resulting Expression

Now that we have simplified the expression inside the parentheses, we can simplify the resulting expression as follows:

$$\left(\frac{3^2 a^{-2}}{3 a^{-1}}\right)^k = \left(\frac{3}{a}\right)^k$$

Step 3: Raise the Simplified Expression to the Power of $k$

Now that we have simplified the expression, we can raise it to the power of $k$ as follows:

$$\left(\frac{3}{a}\right)^k = \frac{3^k}{a^k}$$

Substituting the Given Values

We are given that $a = 5$ and $k = -2$. Substituting these values into the expression, we get:

$$\frac{3^k}{a^k} = \frac{3^{-2}}{5^{-2}} = \frac{\frac{1}{3^2}}{\frac{1}{5^2}} = \frac{\frac{1}{9}}{\frac{1}{25}} = \frac{1}{9} \times \frac{25}{1} = \frac{25}{9}$$

Conclusion

In conclusion, the value of the expression when $a = 5$ and $k = -2$ is $\frac{25}{9}$.

Answer

The correct answer is:

• B. $\frac{9}{25}$ is incorrect
• A. $\frac{1}{75}$ is incorrect
• C. $\frac{25}{9}$ is correct
• D. 75 is incorrect
  Frequently Asked Questions (FAQs) =====================================

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next.
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify exponential expressions?

A: To simplify exponential expressions, we can use the following properties:

• Product of Powers: When we multiply two exponential expressions with the same base, we can add their exponents. For example, $a^m \times a^n = a^{m+n}$.
• Quotient of Powers: When we divide two exponential expressions with the same base, we can subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• Power of a Power: When we raise an exponential expression to a power, we can multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.

Q: How do I evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, we can use the following property:

• Negative Exponent: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Q: How do I raise an expression to a power?

A: To raise an expression to a power, we can use the following property:

• Power of a Product: When we raise a product to a power, we can raise each factor to that power. For example, $(ab)^m = a^m \times b^m$.
• Power of a Quotient: When we raise a quotient to a power, we can raise the numerator and denominator to that power. For example, $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$.

Q: How do I simplify expressions with variables?

A: To simplify expressions with variables, we can use the following properties:

• Like Terms: When we have like terms, we can combine them by adding or subtracting their coefficients. For example, $2x + 3x = 5x$.
• Distributive Property: When we have a term multiplied by a sum, we can distribute the term to each term in the sum. For example, $a(b + c) = ab + ac$.

Q: How do I evaluate expressions with multiple variables?

A: To evaluate expressions with multiple variables, we can use the following properties:

• Substitution: When we have an expression with multiple variables, we can substitute the values of the variables into the expression.
• Order of Operations: When we have multiple operations in an expression, we can use the order of operations (PEMDAS) to evaluate the expression.

Conclusion

In conclusion, simplifying exponential expressions and evaluating expressions with variables and multiple variables requires a strong understanding of the properties of exponents and the order of operations. By following the steps outlined in this article, you can simplify complex expressions and evaluate them with ease.