Which Expression Is Equivalent To { -4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$}$?A. { -4^8$}$ B. { -8^4$}$ C. { (-8)^4$}$ D. { (-4)^8$}$

When dealing with exponents, it's essential to understand the rules and properties that govern them. In this article, we will explore the concept of exponents and how to simplify expressions involving them. We will also examine the given expression {-4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4} and determine which of the provided options is equivalent to it.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, ${4 \times 4 \times 4 \times 4\$} can be written as ${4^4\$}. The exponent tells us how many times the base (in this case, 4) is multiplied by itself.

Properties of Exponents

There are several properties of exponents that we need to be aware of:

• Product of Powers Property: When multiplying two powers with the same base, we add the exponents. For example, ${4^2 \times 4^3 = 4^{2+3} = 4^5\$}.
• Power of a Power Property: When raising a power to another power, we multiply the exponents. For example, {(4²⁾3 = 4^{2 \times 3} = 4^6$}$.
• Negative Exponents: A negative exponent can be rewritten as a positive exponent with a reciprocal. For example, ${4^{-2} = \frac{1}{4^2}\$}.

Evaluating the Given Expression

Now that we have a good understanding of exponents and their properties, let's evaluate the given expression {-4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$}$.

We can start by grouping the 4s together:

{-4 \times (4 \times 4) \times (4 \times 4) \times (4 \times 4) \times (4 \times 4) \times (4 \times 4) \times (4 \times 4)$

Using the product of powers property, we can rewrite this as:

[$-4^8$

Which Option is Equivalent?

Now that we have evaluated the given expression, let's examine the options:

A. [$-4^8\$ B. \[$-8^4$ C. [$(-8)^4\$ D. \[$(-4)^8$

Based on our evaluation, option A is equivalent to the given expression.

Why is Option A Correct?

Option A is correct because it represents the same expression as the given problem. The negative sign is outside the exponent, indicating that the base is negative. The exponent is 8, indicating that the base is multiplied by itself 8 times.

Why are Options B, C, and D Incorrect?

Option B is incorrect because it represents a different expression. The base is 8, not 4, and the exponent is 4, not 8.

Option C is incorrect because it represents a different expression. The base is -8, not -4, and the exponent is 4, not 8.

Option D is incorrect because it represents a different expression. The base is -4, not -4, but the exponent is 8, which is correct. However, the negative sign is inside the exponent, indicating that the base is positive.

Conclusion

In conclusion, the correct answer is option A, [$-4^8\$. This expression is equivalent to the given expression \[$-4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$. We evaluated the expression using the properties of exponents and determined that option A is the correct answer.

Final Thoughts

In the previous article, we explored the concept of exponents and how to simplify expressions involving them. We also evaluated the given expression [-4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\$} and determined which of the provided options is equivalent to it. In this article, we will answer some frequently asked questions about exponents.

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

A: The base is the number being multiplied by itself, and the exponent is the number of times the base is multiplied by itself.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the properties of exponents, such as the product of powers property, the power of a power property, and the negative exponent property.

Q: What is the product of powers property?

A: The product of powers property states that when multiplying two powers with the same base, you add the exponents. For example, ${4^2 \times 4^3 = 4^{2+3} = 4^5\$}.

Q: What is the power of a power property?

A: The power of a power property states that when raising a power to another power, you multiply the exponents. For example, {(4²⁾3 = 4^{2 \times 3} = 4^6$}$.

Q: What is a negative exponent?

A: A negative exponent is a power with a negative exponent. It can be rewritten as a positive exponent with a reciprocal. For example, ${4^{-2} = \frac{1}{4^2}\$}.

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

A: To evaluate an expression with a negative exponent, you can rewrite it as a positive exponent with a reciprocal. For example, ${4^{-2} = \frac{1}{4^2}\$}.

Q: What is the difference between {-4^8$}$ and {(-4)^8$}$?

A: {-4^8$}$ represents a negative base with a positive exponent, while {(-4)^8$}$ represents a positive base with a negative exponent. They are not equivalent expressions.

Q: How do I determine which option is equivalent to a given expression?

A: To determine which option is equivalent to a given expression, you can evaluate the expression using the properties of exponents and compare it to the options.

Q: What are some common mistakes to avoid when working with exponents?

A: Some common mistakes to avoid when working with exponents include:

• Not using the product of powers property when multiplying powers with the same base
• Not using the power of a power property when raising a power to another power
• Not rewriting negative exponents as positive exponents with a reciprocal
• Not evaluating expressions carefully and making mistakes in the calculation

Conclusion

In conclusion, exponents can be a powerful tool for simplifying expressions and solving problems. By understanding the properties of exponents and how to apply them, we can evaluate expressions and determine which option is equivalent. In this article, we answered some frequently asked questions about exponents and provided tips for simplifying expressions and avoiding common mistakes.

Final Thoughts

Exponents are an essential concept in mathematics, and understanding them can help you simplify expressions and solve problems more efficiently. By practicing and applying the properties of exponents, you can become more confident and proficient in working with exponents.