Expressions With Value Of -1/64 A Detailed Evaluation

#h1 Evaluating Expressions to Find -1/64

In the realm of mathematics, particularly when dealing with exponents and fractions, it's crucial to understand how different expressions can yield the same value. This article delves into the process of evaluating expressions involving fractions and exponents, specifically focusing on identifying which expressions result in the value -1/64. This involves understanding the rules of exponents, the impact of negative signs, and how fractions behave when raised to a power. Let's embark on this mathematical journey to dissect each expression and determine its value.

Understanding Exponents and Fractions

When evaluating expressions with exponents and fractions, it's essential to grasp the fundamental principles governing these mathematical operations. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression x^n, 'x' is the base and 'n' is the exponent. This means 'x' is multiplied by itself 'n' times. When dealing with fractions raised to a power, both the numerator and the denominator are raised to that power. For example, (a/b)^n equals a^n / b^n. This understanding forms the bedrock for accurately evaluating the given expressions. Moreover, the sign of the base number plays a pivotal role. A negative number raised to an even power results in a positive number, while a negative number raised to an odd power yields a negative number. These basic rules are indispensable in navigating the intricacies of mathematical expressions.

The interaction between fractions and exponents introduces another layer of complexity. When a fraction is raised to a power, the power is applied to both the numerator and the denominator. Consider the fraction (1/4) raised to the power of 3, denoted as (1/4)^3. This means we need to cube both the numerator (1) and the denominator (4). 1 cubed (1^3) is 1 * 1 * 1 = 1, and 4 cubed (4^3) is 4 * 4 * 4 = 64. Therefore, (1/4)^3 equals 1/64. This principle is crucial for evaluating the given expressions accurately. Understanding how exponents distribute across fractions enables us to simplify and determine the value of each expression, thereby identifying those that equal -1/64. By mastering this concept, we can confidently tackle expressions with fractional bases and exponents.

Furthermore, the presence of a negative sign significantly alters the outcome when combined with exponents. The position of the negative sign is critical. If the negative sign is inside the parentheses, as in (-1/4)^3, it means the entire fraction, including the negative sign, is being raised to the power. In this case, since the exponent is odd (3), the result will be negative. However, if the negative sign is outside the parentheses, as in -(1/4)^3, only the fraction is raised to the power, and the negative sign is applied to the result. This distinction is vital because it affects the sign of the final answer. For instance, (-1/2)^2 will be positive because a negative number squared is positive, whereas -(1/2)^2 will be negative because the square of 1/2 is made negative by the sign outside the parentheses. Recognizing and applying this rule is essential for accurate evaluation and problem-solving.

Analyzing the Expressions

Now, let's dissect each expression to ascertain if it equates to -rac{1}{64}.

1. $\left(-\frac{1}{4}\right)^3$

In this expression, we have a negative fraction, $-\frac{1}{4}$, raised to the power of 3. This signifies that we are multiplying $-\frac{1}{4}$ by itself three times: $\left(-\frac{1}{4}\right) \times \left(-\frac{1}{4}\right) \times \left(-\frac{1}{4}\right)$. When multiplying fractions, we multiply the numerators together and the denominators together. So, we have $(-1) \times (-1) \times (-1)$ in the numerator and $4 \times 4 \times 4$ in the denominator. The product of the numerators is -1, since a negative number multiplied by itself an odd number of times remains negative. The product of the denominators is 64. Therefore, $\left(-\frac{1}{4}\right)^3$ simplifies to $-\frac{1}{64}$. This expression does indeed have the desired value. This evaluation underscores the importance of paying close attention to the sign and exponent when dealing with fractions.

2. $-\left(\frac{1}{4}\right)^3$

Here, we encounter an expression where the negative sign is positioned outside the parentheses, influencing the final result differently. The expression states that we need to first compute $\left(\frac{1}{4}\right)^3$ and then apply the negative sign. Calculating $\left(\frac{1}{4}\right)^3$ involves cubing both the numerator and the denominator: $1^3 = 1$ and $4^3 = 64$. Thus, $\left(\frac{1}{4}\right)^3$ equals $\frac{1}{64}$. Now, applying the negative sign that is outside the parentheses, we get $-\frac{1}{64}$. Consequently, this expression also has the value we are seeking. This example highlights the critical role of the negative sign's placement and its impact on the outcome of the expression. A subtle change in the expression's structure can lead to the same result, emphasizing the importance of meticulous evaluation.

3. $\left(-\frac{1}{8}\right)^2$

In this instance, we are presented with the expression $\left(-\frac{1}{8}\right)^2$, which involves squaring a negative fraction. To evaluate this, we multiply $-\frac{1}{8}$ by itself: $\left(-\frac{1}{8}\right) \times \left(-\frac{1}{8}\right)$. When multiplying fractions, we multiply the numerators and the denominators separately. In this case, the numerator becomes $(-1) \times (-1)$, which equals 1, since a negative number multiplied by a negative number yields a positive result. The denominator becomes $8 \times 8$, which equals 64. Therefore, $\left(-\frac{1}{8}\right)^2$ simplifies to $\frac{1}{64}$. It is crucial to note that this value is positive, not negative, because we are squaring a negative number. As such, this expression does not equal $-\frac{1}{64}$. This example illustrates the significance of the exponent being even, which results in a positive outcome when applied to a negative base.

4. $-\left(\frac{1}{8}\right)^2$

This expression, $-\left(\frac{1}{8}\right)^2$, presents a scenario where we first square the fraction $\frac{1}{8}$ and then apply the negative sign. Squaring $\frac{1}{8}$ means calculating $\left(\frac{1}{8}\right) \times \left(\frac{1}{8}\right)$. Multiplying the numerators gives $1 \times 1 = 1$, and multiplying the denominators gives $8 \times 8 = 64$. So, $\left(\frac{1}{8}\right)^2$ equals $\frac{1}{64}$. Next, we apply the negative sign that is outside the parentheses, resulting in $-\frac{1}{64}$. Consequently, this expression does indeed have the value we are looking for. This example reinforces the importance of adhering to the order of operations, where exponentiation precedes negation. The placement of the negative sign relative to the parentheses plays a pivotal role in determining the final value of the expression.

5. $\left(-\frac{1}{2}\right)^6$

Finally, we examine the expression $\left(-\frac{1}{2}\right)^6$, where a negative fraction is raised to the power of 6. This means we multiply $-\frac{1}{2}$ by itself six times. Since the exponent is even, the negative sign will cancel out, resulting in a positive value. To calculate the value, we raise both the numerator and the denominator to the power of 6: $(-1)^6 = 1$ and $2^6 = 64$. Therefore, $\left(-\frac{1}{2}\right)^6$ simplifies to $\frac{1}{64}$. This result is positive, which means this expression does not equal $-\frac{1}{64}$. This final example further illustrates the rule that a negative number raised to an even power yields a positive result, a key concept in understanding exponents and signs.

Conclusion

In summary, after evaluating the given expressions, we find that $\left(-\frac{1}{4}\right)^3$ and $-\left(\frac{1}{4}\right)^3$ and $-\left(\frac{1}{8}\right)^2$ are the expressions that have a value of $-\frac{1}{64}$. This exercise underscores the importance of understanding the rules of exponents, the impact of negative signs, and how fractions behave when raised to a power. By meticulously applying these principles, we can accurately evaluate mathematical expressions and solve problems involving fractions and exponents. The careful consideration of each component, from the exponent to the sign, ensures precision in mathematical computations.

#h2 Key Takeaways

• Exponents and Fractions: When a fraction is raised to a power, both the numerator and denominator are raised to that power.
• Negative Signs: A negative number raised to an odd power results in a negative number, while a negative number raised to an even power results in a positive number.
• Order of Operations: Pay close attention to the order of operations, particularly when negative signs and exponents are involved.
• Careful Evaluation: Meticulously evaluate each expression, considering the placement of parentheses and negative signs.

#h3 Further Exploration

To deepen your understanding of these concepts, consider exploring additional examples and practice problems. You can also investigate more complex expressions involving fractional exponents and negative exponents. Consistent practice and exploration will solidify your grasp of these fundamental mathematical principles.