Which Shows How To Find The Value Of This Expression When $x = -2$ And $y = 5$? ( 3 X 3 Y − 2 ) 2 \left(3 X^3 Y^{-2}\right)^2 ( 3 X 3 Y − 2 ) 2 A. 3 2 ( − 2 ) 6 5 4 \frac{3^2(-2)^6}{5^4} 5 4 3 2 ( − 2 ) 6 ​ B. 3 ( − 2 ) 8 5 4 \frac{3(-2)^8}{5^4} 5 4 3 ( − 2 ) 8 ​ C. 3 2 ( 5 ) 6 ( − 2 ) 4 \frac{3^2(5)^6}{(-2)^4} ( − 2 ) 4 3 2 ( 5 ) 6 ​ D.

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Introduction

In mathematics, exponential expressions are a fundamental concept that plays a crucial role in various mathematical operations. When dealing with exponential expressions, it's essential to understand the rules and properties that govern their behavior. In this article, we will explore how to find the value of a given expression when the variables x and y are assigned specific values.

Understanding Exponential Expressions

Exponential expressions are mathematical expressions that involve the exponentiation of a base number. The general form of an exponential expression is:

aba^b

where a is the base number and b is the exponent. When dealing with exponential expressions, it's essential to understand the rules of exponentiation, including the product rule, power rule, and quotient rule.

The Product Rule

The product rule states that when multiplying two exponential expressions with the same base, the exponents are added together. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

The Power Rule

The power rule states that when raising an exponential expression to a power, the exponents are multiplied together. Mathematically, this can be represented as:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

The Quotient Rule

The quotient rule states that when dividing two exponential expressions with the same base, the exponents are subtracted. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

Solving the Given Expression

Now that we have a solid understanding of the rules of exponentiation, let's apply them to the given expression:

(3x3y−2)2\left(3 x^3 y^{-2}\right)^2

To find the value of this expression when x = -2 and y = 5, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses:

3x3y−23 x^3 y^{-2}

  1. Raise the result to the power of 2:

(3x3y−2)2\left(3 x^3 y^{-2}\right)^2

Step 1: Evaluate the Expression Inside the Parentheses

Using the product rule, we can rewrite the expression as:

3x3y−2=3⋅x3⋅y−23 x^3 y^{-2} = 3 \cdot x^3 \cdot y^{-2}

Now, let's substitute the values of x and y:

x=−2x = -2

y=5y = 5

Substituting these values into the expression, we get:

3⋅(−2)3⋅5−23 \cdot (-2)^3 \cdot 5^{-2}

Step 2: Raise the Result to the Power of 2

Now that we have evaluated the expression inside the parentheses, let's raise the result to the power of 2:

(3⋅(−2)3⋅5−2)2\left(3 \cdot (-2)^3 \cdot 5^{-2}\right)^2

Using the power rule, we can rewrite the expression as:

32⋅(−2)6⋅5−43^2 \cdot (-2)^6 \cdot 5^{-4}

Simplifying the Expression

Now that we have raised the result to the power of 2, let's simplify the expression:

32⋅(−2)6⋅5−4=32(−2)6543^2 \cdot (-2)^6 \cdot 5^{-4} = \frac{3^2(-2)^6}{5^4}

Conclusion

In this article, we have explored how to find the value of a given expression when the variables x and y are assigned specific values. We have applied the rules of exponentiation, including the product rule, power rule, and quotient rule, to simplify the expression. The final answer is:

32(−2)654\frac{3^2(-2)^6}{5^4}

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is a mathematical expression that involves the exponentiation of a base number, whereas a polynomial expression is a mathematical expression that involves the sum of terms with variables raised to non-negative integer powers.

Q: What is the product rule for exponential expressions?

A: The product rule states that when multiplying two exponential expressions with the same base, the exponents are added together. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

Q: What is the power rule for exponential expressions?

A: The power rule states that when raising an exponential expression to a power, the exponents are multiplied together. Mathematically, this can be represented as:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

Q: What is the quotient rule for exponential expressions?

A: The quotient rule states that when dividing two exponential expressions with the same base, the exponents are subtracted. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can use the rules of exponentiation, including the product rule, power rule, and quotient rule. You can also use the properties of exponents, such as the fact that am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the order of operations for exponential expressions?

A: The order of operations for exponential expressions is:

  1. Evaluate the expression inside the parentheses
  2. Raise the result to the power of 2
  3. Simplify the expression using the rules of exponentiation

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

A: To evaluate an exponential expression with a negative exponent, you can use the fact that a−n=1ana^{-n} = \frac{1}{a^n}. For example, if you have the expression a−2a^{-2}, you can rewrite it as 1a2\frac{1}{a^2}.

Q: What is the difference between an exponential expression and a logarithmic expression?

A: An exponential expression is a mathematical expression that involves the exponentiation of a base number, whereas a logarithmic expression is a mathematical expression that involves the inverse operation of exponentiation. In other words, a logarithmic expression is the inverse of an exponential expression.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the fact that if am=ana^m = a^n, then m=nm = n. You can also use the properties of exponents, such as the fact that am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the significance of exponential expressions in real-world applications?

A: Exponential expressions have many real-world applications, including population growth, compound interest, and chemical reactions. They are also used in many fields, such as physics, engineering, and economics.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponential expressions. We have covered topics such as the product rule, power rule, and quotient rule, as well as the order of operations and how to simplify exponential expressions. We hope that this article has been helpful in understanding exponential expressions and their applications.