Select The Correct Answer.Which Expression Is Equivalent To The Given Expression? $\left(2 W^{-2}\right)^3\left(8 W^6\right$\]A. 48 B. 64 C. $\frac{64}{w^2}$ D. $\frac{48}{w^2}$

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore how to simplify exponential expressions, with a focus on the given expression $\left(2 w^{-2}\right)^3\left(8 w^6\right)$. We will break down the solution step by step, using the properties of exponents to arrive at the correct answer.

Understanding Exponents

Before we dive into the solution, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number. It indicates how many times the base number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2$.

Simplifying the Given Expression

Now, let's simplify the given expression $\left(2 w^{-2}\right)^3\left(8 w^6\right)$. To do this, we will use the properties of exponents, specifically the power of a power property and the product of powers property.

Step 1: Apply the Power of a Power Property

The power of a power property states that when we raise a power to a power, we multiply the exponents. In this case, we have $\left(2 w^{-2}\right)^3$. Using the power of a power property, we can rewrite this as $2^3 \left(w^{-2}\right)^3$.

import sympy as sp

# Define the variables
w = sp.symbols('w')

# Apply the power of a power property
expr = (2 * w**(-2))**3
simplified_expr = 2**3 * (w**(-2))**3
print(simplified_expr)

Step 2: Simplify the Exponents

Now, let's simplify the exponents. We have $2^3 \left(w^{-2}\right)^3$. Using the property of exponents that states $\left(a^m\right)^n = a^{mn}$, we can rewrite this as $2^3 w^{-6}$.

# Simplify the exponents
simplified_expr = 2**3 * w**(-6)
print(simplified_expr)

Step 3: Multiply the Terms

Now, let's multiply the terms. We have $2^3 w^{-6} \left(8 w^6\right)$. Using the product of powers property, we can rewrite this as $2^3 \cdot 8 \cdot w^{-6} \cdot w^6$.

# Multiply the terms
simplified_expr = 2**3 * 8 * w**(-6) * w**6
print(simplified_expr)

Step 4: Simplify the Expression

Now, let's simplify the expression. We have $2^3 \cdot 8 \cdot w^{-6} \cdot w^6$. Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite this as $2^3 \cdot 8 \cdot w^{-6+6}$.

# Simplify the expression
simplified_expr = 2**3 * 8 * w**(-6+6)
print(simplified_expr)

Step 5: Evaluate the Expression

Finally, let's evaluate the expression. We have $2^3 \cdot 8 \cdot w^{-6+6}$. Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite this as $2^3 \cdot 8 \cdot w^0$.

# Evaluate the expression
simplified_expr = 2**3 * 8 * w**0
print(simplified_expr)

Step 6: Simplify the Final Expression

Now, let's simplify the final expression. We have $2^3 \cdot 8 \cdot w^0$. Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite this as $2^3 \cdot 8 \cdot 1$.

# Simplify the final expression
simplified_expr = 2**3 * 8 * 1
print(simplified_expr)

Step 7: Evaluate the Final Expression

Finally, let's evaluate the final expression. We have $2^3 \cdot 8 \cdot 1$. Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite this as $64$.

# Evaluate the final expression
simplified_expr = 64
print(simplified_expr)

Conclusion

In this article, we simplified the given expression $\left(2 w^{-2}\right)^3\left(8 w^6\right)$ using the properties of exponents. We applied the power of a power property, simplified the exponents, multiplied the terms, simplified the expression, evaluated the expression, and finally simplified the final expression to arrive at the correct answer, which is $\boxed{64}$.

Discussion

The given expression $\left(2 w^{-2}\right)^3\left(8 w^6\right)$ is a classic example of how to simplify exponential expressions using the properties of exponents. The power of a power property, the product of powers property, and the property of exponents that states $a^m \cdot a^n = a^{m+n}$ are all essential tools to master when working with exponential expressions.

Practice Problems

Here are some practice problems to help you master the skills learned in this article:

1. Simplify the expression $\left(3 x^2\right)^4\left(9 x^6\right)$.
2. Simplify the expression $\left(2 y^{-3}\right)^2\left(8 y^9\right)$.
3. Simplify the expression $\left(4 z^4\right)^3\left(16 z^12\right)$.

Answer Key

1. $\boxed{81 x^{16}}$
2. $\boxed{64 y^6}$
3. $\boxed{4096 z^{40}}$

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

About the Author

Introduction

In our previous article, we explored how to simplify exponential expressions using the properties of exponents. In this article, we will answer some frequently asked questions about simplifying exponential expressions.

Q: What is the power of a power property?

A: The power of a power property states that when we raise a power to a power, we multiply the exponents. For example, $\left(2^3\right)^4 = 2^{3 \cdot 4} = 2^{12}$.

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

A: To simplify an expression with a negative exponent, we can use the property that states $a^{-m} = \frac{1}{a^m}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two powers with the same base, we add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

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

A: To simplify an expression with a zero exponent, we can use the property that states $a^0 = 1$. For example, $2^0 = 1$.

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This property states that when we multiply two powers with the same base, we add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

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

A: To simplify an expression with a fractional exponent, we can use the property that states $a^{m/n} = \sqrt[n]{a^m}$. For example, $2^{3/4} = \sqrt[4]{2^3} = \sqrt[4]{8}$.

Q: What is the property of exponents that states $\left(a^m\right)^n = a^{mn}$?

A: This property states that when we raise a power to a power, we multiply the exponents. For example, $\left(2^3\right)^4 = 2^{3 \cdot 4} = 2^{12}$.

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

A: To simplify an expression with a negative fractional exponent, we can use the property that states $a^{-m/n} = \frac{1}{a^{m/n}}$. For example, $2^{-3/4} = \frac{1}{2^{3/4}} = \frac{1}{\sqrt[4]{8}}$.

Conclusion

In this article, we answered some frequently asked questions about simplifying exponential expressions. We covered topics such as the power of a power property, the product of powers property, and the property of exponents that states $a^m \cdot a^n = a^{m+n}$. We also discussed how to simplify expressions with negative exponents, zero exponents, fractional exponents, and negative fractional exponents.

Practice Problems

Here are some practice problems to help you master the skills learned in this article:

1. Simplify the expression $\left(3 x^2\right)^4\left(9 x^6\right)$.
2. Simplify the expression $\left(2 y^{-3}\right)^2\left(8 y^9\right)$.
3. Simplify the expression $\left(4 z^4\right)^3\left(16 z^{12}\right)$.

Answer Key

1. $\boxed{81 x^{16}}$
2. $\boxed{64 y^6}$
3. $\boxed{4096 z^{40}}$

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

About the Author

The author is a mathematics educator with over 10 years of experience teaching algebra, trigonometry, and calculus. They have a passion for making mathematics accessible and enjoyable for students of all ages and skill levels.