Study The Expression: { \left(4 M^ -3} N 2\right) 2$}$What Is The Value Of The Expression When { M = 2$}$ And { N = -3$}$?1. Apply The Power Of A Power ${4^2 M^{-6 N^4}$2. Substitute Values For Variables:

Mar 10, 2025 by ADMIN 206 views

**Simplifying Exponential Expressions: A Step-by-Step Guide**

Understanding the Expression

The given expression is ${$ \left(4 m^{-3} n^2\right)2$}$. This expression involves exponents and variables, and we are asked to find its value when ${$ m = 2$}$ and ${$ n = -3$}$. To simplify this expression, we will apply the power of a power rule, which states that for any variables a and b and any integer n, ${(ab)^n = a^nb^n}$ .

Applying the Power of a Power

To simplify the given expression, we will apply the power of a power rule. This rule allows us to distribute the exponent outside the parentheses to each variable inside. In this case, we have:

${$ \left(4 m^{-3} n^2\right)2 = 4^2 m^{-6} n^4$}$

This simplifies the expression by distributing the exponent 2 to each variable inside the parentheses.

Substituting Values for Variables

Now that we have simplified the expression, we can substitute the given values for the variables m and n. We are given that ${$ m = 2$}$ and ${$ n = -3$}$. Substituting these values into the simplified expression, we get:

${ $4^2 (2)^{-6} (-3)^4\$}$

To evaluate this expression, we need to follow the order of operations (PEMDAS):

Evaluate the exponent 2: ${ $4^2 = 16\$}$
Evaluate the exponent -6: ${$ (2)^{-6} = \frac{1}{2^6} = \frac{1}{64}$}$
Evaluate the exponent 4: ${$ (-3)^4 = 81$}$

Now that we have evaluated the exponents, we can substitute these values back into the expression:

${ $16 \times \frac{1}{64} \times 81\$}$

To simplify this expression, we can multiply the numerators and denominators:

${$ \frac{16 \times 81}{64} = \frac{1296}{64} = 20.25$}$

Therefore, the value of the expression when ${$ m = 2$}$ and ${$ n = -3$}$ is 20.25.

Conclusion

In this article, we simplified the expression ${$ \left(4 m^{-3} n^2\right)2$}$ by applying the power of a power rule. We then substituted the given values for the variables m and n and evaluated the expression using the order of operations. The final value of the expression is 20.25.

Key Takeaways

The power of a power rule allows us to distribute the exponent outside the parentheses to each variable inside.
To simplify an expression with exponents, we need to follow the order of operations (PEMDAS).
Substituting values for variables is an important step in evaluating an expression.

Discussion Questions

What is the power of a power rule, and how is it used to simplify expressions?
How do you evaluate an expression with exponents using the order of operations (PEMDAS)?
What is the final value of the expression ${$ \left(4 m^{-3} n^2\right)2$}$ when ${$ m = 2$}$ and ${$ n = -3$}$?
Frequently Asked Questions: Simplifying Exponential Expressions ================================================================

Q: What is the power of a power rule, and how is it used to simplify expressions?

A: The power of a power rule is a mathematical rule that allows us to distribute the exponent outside the parentheses to each variable inside. This rule is used to simplify expressions with exponents by making it easier to evaluate the expression.

Q: How do you evaluate an expression with exponents using the order of operations (PEMDAS)?

A: To evaluate an expression with exponents using the order of operations (PEMDAS), you need to follow these steps:

Evaluate the exponents: Evaluate any exponents in the expression, such as 2^3 or (-2)^2.
Multiply and divide from left to right: Multiply and divide any numbers in the expression from left to right.
Add and subtract from left to right: Add and subtract any numbers in the expression from left to right.

Q: What is the final value of the expression ${$ \left(4 m^{-3} n^2\right)2$}$ when ${$ m = 2$}$ and ${$ n = -3$}$?

A: To find the final value of the expression, we need to substitute the given values for the variables m and n into the expression and evaluate it using the order of operations (PEMDAS). The final value of the expression is 20.25.

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

A: To simplify an expression with multiple exponents, you need to apply the power of a power rule to each exponent separately. For example, if you have the expression ${$ (2^3 \times 3²⁾2$}$, you would first simplify the expression inside the parentheses using the power of a power rule, and then evaluate the resulting expression.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example, ${ $2^3 = 8\$}$ and ${ $2^{-3} = \frac{1}{2^3} = \frac{1}{8}\$}$ .

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

A: To evaluate an expression with a negative exponent, you need to take the reciprocal of the base and then raise it to the power of the absolute value of the exponent. For example, ${ $2^{-3} = \frac{1}{2^3} = \frac{1}{8}\$}$ .

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

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

P: Parentheses
E: Exponents
M: Multiplication
D: Division
A: Addition
S: Subtraction

Q: Why is it important to follow the order of operations (PEMDAS)?

A: Following the order of operations (PEMDAS) is important because it ensures that mathematical expressions are evaluated consistently and accurately. If the order of operations is not followed, the result of the expression may be incorrect.

Q: Can you provide examples of expressions that require the use of the power of a power rule?

A: Yes, here are a few examples of expressions that require the use of the power of a power rule:

[$(2^3 \times 3²⁾2$
[$(4^2 \times 5³⁾4$
[$(6^2 \times 7³⁾5$

These expressions require the use of the power of a power rule to simplify and evaluate them.