Evaluate The Expression ( 2 X 17 Y ) 3 \left(2 X^{17} Y\right)^3 ( 2 X 17 Y ) 3 For X = − 1 X=-1 X = − 1 And Y = 5 Y=5 Y = 5 .A. -10 B. -250 C. -1,000 D. -1,530 Please Select The Best Answer From The Choices Provided: A B C D

Introduction

In mathematics, expressions with exponents are a fundamental concept that can be used to simplify complex calculations. When evaluating expressions with exponents, it's essential to understand the rules of exponentiation and how to apply them to different scenarios. In this article, we'll explore how to evaluate the expression $\left(2 x^{17} y\right)^3$ for $x=-1$ and $y=5$.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ can be read as "x to the power of 3" and is equivalent to $x \times x \times x$. When evaluating expressions with exponents, it's crucial 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 numbers with the same base, you can add their exponents. For example, $x^2 \times x^3 = x^{2+3} = x^5$. This rule can be applied to expressions with multiple terms, such as $\left(2 x^{17} y\right)^3$.

The Power Rule

The power rule states that when raising a power to another power, you can multiply the exponents. For example, $\left(x^2\right)^3 = x^{2 \times 3} = x^6$. This rule can be applied to expressions with exponents, such as $\left(2 x^{17} y\right)^3$.

The Quotient Rule

The quotient rule states that when dividing two numbers with the same base, you can subtract their exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$. This rule can be applied to expressions with fractions, such as $\frac{\left(2 x^{17} y\right)^3}{x^2}$.

Evaluating the Expression

Now that we've reviewed the rules of exponentiation, let's evaluate the expression $\left(2 x^{17} y\right)^3$ for $x=-1$ and $y=5$. To do this, we'll apply the product rule and power rule.

Step 1: Apply the Product Rule

Using the product rule, we can rewrite the expression as $2^3 \times \left(x^{17}\right)^3 \times y^3$. This simplifies to $8 \times x^{51} \times y^3$.

Step 2: Substitute the Values of x and y

Now that we have the simplified expression, we can substitute the values of $x$ and $y$. We're given that $x=-1$ and $y=5$, so we can substitute these values into the expression.

Step 3: Evaluate the Expression

Substituting the values of $x$ and $y$, we get $8 \times (-1)^{51} \times 5^3$. To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent $(-1)^{51}$.
2. Evaluate the exponent $5^3$.
3. Multiply the results.

**Step 4: Evaluate the Exponent $(-1)^{51}$

The exponent $(-1)^{51}$ can be evaluated by following the rules of exponentiation. Since $(-1)^{51}$ is an odd exponent, the result is $-1$.

**Step 5: Evaluate the Exponent $5^3$

The exponent $5^3$ can be evaluated by multiplying 5 by itself three times: $5 \times 5 \times 5 = 125$.

Step 6: Multiply the Results

Now that we have the results of the exponents, we can multiply them together: $8 \times (-1) \times 125 = -1000$.

Conclusion

In this article, we evaluated the expression $\left(2 x^{17} y\right)^3$ for $x=-1$ and $y=5$. By applying the product rule and power rule, we simplified the expression to $8 \times x^{51} \times y^3$. Substituting the values of $x$ and $y$, we evaluated the expression to get $-1000$. Therefore, the correct answer is:

Introduction

In our previous article, we explored how to evaluate the expression $\left(2 x^{17} y\right)^3$ for $x=-1$ and $y=5$. In this article, we'll answer some frequently asked questions about evaluating expressions with exponents.

Q: What is the order of operations when evaluating expressions with exponents?

A: When evaluating expressions with exponents, it's essential to follow the order of operations (PEMDAS):

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

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

A: When evaluating an expression with multiple exponents, you can use the product rule and power rule to simplify the expression. For example, consider the expression $\left(2 x^{17} y\right)^3$. Using the product rule, we can rewrite the expression as $2^3 \times \left(x^{17}\right)^3 \times y^3$. This simplifies to $8 \times x^{51} \times y^3$.

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, $x^3$ is equivalent to $x \times x \times x$, while $x^{-3}$ is equivalent to $\frac{1}{x \times x \times x}$.

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

A: When evaluating an expression with a negative exponent, you can use the rule that $x^{-n} = \frac{1}{x^n}$. For example, consider the expression $\frac{1}{x^3}$. Using this rule, we can rewrite the expression as $x^{-3}$.

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is raised to a power, while a power is the result of raising a number to an exponent. For example, $x^3$ is an exponent, while $x^3 = 8$ is a power.

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

A: When evaluating an expression with a fractional exponent, you can use the rule that $x^{m/n} = \sqrt[n]{x^m}$. For example, consider the expression $\sqrt[3]{x^2}$. Using this rule, we can rewrite the expression as $x^{2/3}$.

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

A: An exponential function is a function that involves an exponent, while a polynomial function is a function that involves only addition, subtraction, multiplication, and division. For example, $f(x) = 2^x$ is an exponential function, while $f(x) = x^2 + 3x - 4$ is a polynomial function.

Conclusion

In this article, we answered some frequently asked questions about evaluating expressions with exponents. We covered topics such as the order of operations, evaluating expressions with multiple exponents, positive and negative exponents, and fractional exponents. By understanding these concepts, you'll be better equipped to evaluate expressions with exponents and solve problems involving exponential functions.

Additional Resources

If you're looking for more information on evaluating expressions with exponents, here are some additional resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Practice Problems

Try these practice problems to test your understanding of evaluating expressions with exponents:

1. Evaluate the expression $\left(3 x^2 y\right)^4$ for $x=2$ and $y=3$.
2. Evaluate the expression $\left(2 x^{-3} y\right)^2$ for $x=-1$ and $y=5$.
3. Evaluate the expression $\sqrt[3]{x^2}$ for $x=8$.
4. Evaluate the expression $x^{2/3}$ for $x=27$.

Answer Key

1. $81 \times 2^8 \times 3^4 = 104652$
2. $4 \times 2^6 \times 5^2 = 20000$
3. $4$
4. $9$