Which Expressions Are Equivalent To -64? Check TWO That Apply.A. $4^{-3}$ B. $2^3 \cdot 2^2 \cdot 2^0$ C. $-2^{-2} \cdot (2^8$\] D. $-\left(\frac{1}{4}\right)^{-3}$ E. $\frac{2^8 - 2^4}{2^2}$

In mathematics, equivalent expressions are those that have the same value, even if they are written differently. In this article, we will explore five different expressions and determine which two are equivalent to -64.

Understanding Exponents

Before we dive into the expressions, let's review some basic exponent rules. An exponent is a small number that is raised to the power of a larger number. For example, in the expression $2^3$, the 3 is the exponent and the 2 is the base. When we multiply the base by itself as many times as the exponent tells us, we get the result. In this case, $2^3 = 2 \cdot 2 \cdot 2 = 8$.

Expression A: $4^{-3}$

The first expression we will examine is $4^{-3}$. To evaluate this expression, we need to understand that a negative exponent means we are taking the reciprocal of the base raised to the positive exponent. In this case, $4^{-3} = \frac{1}{4^3}$. We can simplify this expression by evaluating the exponent: $4^3 = 4 \cdot 4 \cdot 4 = 64$. Therefore, $4^{-3} = \frac{1}{64}$.

Expression B: $2^3 \cdot 2^2 \cdot 2^0$

The second expression we will examine is $2^3 \cdot 2^2 \cdot 2^0$. To evaluate this expression, we need to understand that when we multiply numbers with the same base, we can add the exponents. In this case, $2^3 \cdot 2^2 \cdot 2^0 = 2^{3+2+0} = 2^5$. We can simplify this expression by evaluating the exponent: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$.

Expression C: $-2^{-2} \cdot (2^8)$

The third expression we will examine is $-2^{-2} \cdot (2^8)$. To evaluate this expression, we need to understand that a negative exponent means we are taking the reciprocal of the base raised to the positive exponent. In this case, $-2^{-2} = -\frac{1}{2^2}$. We can simplify this expression by evaluating the exponent: $2^2 = 2 \cdot 2 = 4$. Therefore, $-2^{-2} = -\frac{1}{4}$. We can then multiply this expression by $2^8$: $-\frac{1}{4} \cdot 2^8 = -\frac{1}{4} \cdot 256 = -64$.

Expression D: $-\left(\frac{1}{4}\right)^{-3}$

The fourth expression we will examine is $-\left(\frac{1}{4}\right)^{-3}$. To evaluate this expression, we need to understand that a negative exponent means we are taking the reciprocal of the base raised to the positive exponent. In this case, $\left(\frac{1}{4}\right)^{-3} = \left(\frac{4}{1}\right)^3$. We can simplify this expression by evaluating the exponent: $\left(\frac{4}{1}\right)^3 = \frac{4^3}{1^3} = \frac{64}{1} = 64$. Therefore, $-\left(\frac{1}{4}\right)^{-3} = -64$.

Expression E: $\frac{2^8 - 2^4}{2^2}$

The fifth expression we will examine is $\frac{2^8 - 2^4}{2^2}$. To evaluate this expression, we need to understand that when we subtract numbers with the same base, we can subtract the exponents. In this case, $2^8 - 2^4 = 2^{8-4} = 2^4$. We can then divide this expression by $2^2$: $\frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4$.

Conclusion

In conclusion, we have examined five different expressions and determined which two are equivalent to -64. The two expressions that are equivalent to -64 are:

• Expression C: $-2^{-2} \cdot (2^8)$: This expression is equivalent to -64 because $-2^{-2} = -\frac{1}{4}$ and $-\frac{1}{4} \cdot 2^8 = -64$.
• Expression D: $-\left(\frac{1}{4}\right)^{-3}$: This expression is equivalent to -64 because $\left(\frac{1}{4}\right)^{-3} = 64$ and $-\left(\frac{1}{4}\right)^{-3} = -64$.

In this article, we will answer some of the most frequently asked questions related to the expressions equivalent to -64.

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

A: A positive exponent means we are multiplying the base by itself as many times as the exponent tells us. For example, in the expression $2^3$, the 3 is the exponent and the 2 is the base. When we multiply the base by itself as many times as the exponent tells us, we get the result. In this case, $2^3 = 2 \cdot 2 \cdot 2 = 8$. A negative exponent means we are taking the reciprocal of the base raised to the positive exponent. For example, in the expression $4^{-3}$, the -3 is the exponent and the 4 is the base. When we take the reciprocal of the base raised to the positive exponent, we get the result. In this case, $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, we need to understand the rules of exponents. When we multiply numbers with the same base, we can add the exponents. For example, in the expression $2^3 \cdot 2^2$, we can add the exponents: $2^3 \cdot 2^2 = 2^{3+2} = 2^5$. We can then evaluate the exponent: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$. When we divide numbers with the same base, we can subtract the exponents. For example, in the expression $\frac{2^8}{2^2}$, we can subtract the exponents: $\frac{2^8}{2^2} = 2^{8-2} = 2^6$. We can then evaluate the exponent: $2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$.

Q: What is the difference between a reciprocal and a fraction?

A: A reciprocal is the inverse of a number. For example, the reciprocal of 4 is $\frac{1}{4}$. A fraction is a way of expressing a part of a whole. For example, the fraction $\frac{1}{4}$ represents one part out of four equal parts. In the expression $4^{-3}$, we are taking the reciprocal of the base raised to the positive exponent. In this case, $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$.

Q: How do I evaluate an expression with parentheses?

A: To evaluate an expression with parentheses, we need to follow the order of operations (PEMDAS). PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. When we have parentheses in an expression, we need to evaluate the expression inside the parentheses first. For example, in the expression $-2^{-2} \cdot (2^8)$, we need to evaluate the expression inside the parentheses first: $2^8 = 256$. We can then multiply this expression by $-2^{-2}$: $-2^{-2} \cdot 256 = -\frac{1}{4} \cdot 256 = -64$.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example, in the expression $x^2$, the $x$ is a variable. A constant is a value that does not change. For example, in the expression $2^3$, the 2 is a constant. In the expression $-\left(\frac{1}{4}\right)^{-3}$, the $\frac{1}{4}$ is a constant.

Q: How do I simplify an expression with variables and constants?

A: To simplify an expression with variables and constants, we need to understand the rules of exponents. When we multiply numbers with the same base, we can add the exponents. For example, in the expression $2x^2 \cdot 2x$, we can add the exponents: $2x^2 \cdot 2x = 2 \cdot 2 \cdot x^2 \cdot x = 4x^3$. When we divide numbers with the same base, we can subtract the exponents. For example, in the expression $\frac{2x^2}{2x}$, we can subtract the exponents: $\frac{2x^2}{2x} = x^{2-1} = x$.

Conclusion

In conclusion, we have answered some of the most frequently asked questions related to the expressions equivalent to -64. We hope this article has helped you understand the concepts of exponents, reciprocals, fractions, and variables and constants. If you have any further questions or need further clarification, please don't hesitate to ask.