Which Expression Simplifies To Equal − 1 2 -\frac{1}{2} − 2 1 ​ ?A. − 2 2 2 -2 \sqrt{2^2} − 2 2 2 ​ B. 2 − 1 64 3 2 \sqrt[3]{\frac{-1}{64}} 2 3 64 − 1 ​ ​ C. − 1 ⋅ 2 3 -1 \cdot \sqrt{2^3} − 1 ⋅ 2 3 ​ D. − 8 3 \sqrt[3]{-8} 3 − 8 ​

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with radicals and exponents, it's essential to understand the properties and rules that govern their behavior. In this article, we will explore four different expressions and determine which one simplifies to equal −12-\frac{1}{2}.

Understanding the Options

Before we dive into the simplification process, let's take a closer look at the options provided:

  • A. −222-2 \sqrt{2^2}
  • B. 2−16432 \sqrt[3]{\frac{-1}{64}}
  • C. −1⋅23-1 \cdot \sqrt{2^3}
  • D. −83\sqrt[3]{-8}

Each option involves a combination of radicals, exponents, and coefficients. To simplify these expressions, we need to apply the rules of exponents and radicals.

Simplifying Option A

Let's start with option A: −222-2 \sqrt{2^2}. To simplify this expression, we need to evaluate the exponent inside the square root. Since 22=42^2 = 4, we can rewrite the expression as:

−24-2 \sqrt{4}

Now, we can simplify the square root by taking the square root of 4, which is 2:

−2⋅2-2 \cdot 2

Multiplying the coefficient -2 by 2 gives us:

−4-4

However, this is not equal to −12-\frac{1}{2}. Let's move on to the next option.

Simplifying Option B

Now, let's consider option B: 2−16432 \sqrt[3]{\frac{-1}{64}}. To simplify this expression, we need to evaluate the cube root inside the radical. Since −164=−143\frac{-1}{64} = \frac{-1}{4^3}, we can rewrite the expression as:

2−14332 \sqrt[3]{\frac{-1}{4^3}}

Now, we can simplify the cube root by taking the cube root of −143\frac{-1}{4^3}, which is −14\frac{-1}{4}:

2⋅−142 \cdot \frac{-1}{4}

Multiplying the coefficient 2 by −14\frac{-1}{4} gives us:

−12\frac{-1}{2}

This expression is equal to −12-\frac{1}{2}.

Simplifying Option C

Now, let's consider option C: −1⋅23-1 \cdot \sqrt{2^3}. To simplify this expression, we need to evaluate the exponent inside the square root. Since 23=82^3 = 8, we can rewrite the expression as:

−1⋅8-1 \cdot \sqrt{8}

Now, we can simplify the square root by taking the square root of 8, which is 4⋅2=22\sqrt{4 \cdot 2} = 2\sqrt{2}:

−1⋅22-1 \cdot 2\sqrt{2}

Multiplying the coefficient -1 by 222\sqrt{2} gives us:

−22-2\sqrt{2}

However, this is not equal to −12-\frac{1}{2}. Let's move on to the next option.

Simplifying Option D

Now, let's consider option D: −83\sqrt[3]{-8}. To simplify this expression, we need to evaluate the cube root. Since −8=(−2)3-8 = (-2)^3, we can rewrite the expression as:

(−2)33\sqrt[3]{(-2)^3}

Now, we can simplify the cube root by taking the cube root of (−2)3(-2)^3, which is -2:

−2-2

However, this is not equal to −12-\frac{1}{2}. Let's summarize our findings.

Conclusion

In this article, we explored four different expressions and determined which one simplifies to equal −12-\frac{1}{2}. We found that option B: 2−16432 \sqrt[3]{\frac{-1}{64}} is the correct answer. This expression simplifies to −12\frac{-1}{2}, which is equal to −12-\frac{1}{2}.

Final Answer

The final answer is option B: 2−16432 \sqrt[3]{\frac{-1}{64}}.

Introduction

In our previous article, we explored four different expressions and determined which one simplifies to equal −12-\frac{1}{2}. We also discussed the rules and properties of exponents and radicals. In this article, we will answer some frequently asked questions (FAQs) about simplifying expressions.

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

A: A radical is a mathematical operation that involves taking the square root or cube root of a number, while an exponent is a mathematical operation that involves raising a number to a power. For example, 4\sqrt{4} is a radical, while 424^2 is an exponent.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to evaluate the expression inside the radical and then simplify the result. For example, 16\sqrt{16} can be simplified as 4⋅4=4\sqrt{4 \cdot 4} = 4.

Q: What is the rule for simplifying a cube root?

A: The rule for simplifying a cube root is to take the cube root of the number inside the radical and then simplify the result. For example, 643\sqrt[3]{64} can be simplified as 433=4\sqrt[3]{4^3} = 4.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to evaluate each radical separately and then combine the results. For example, 16+9\sqrt{16} + \sqrt{9} can be simplified as 4+3=74 + 3 = 7.

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

A: A positive exponent means that the base is raised to a power, while a negative exponent means that the base is taken to the reciprocal of the power. For example, 424^2 is a positive exponent, while 4−24^{-2} is a negative exponent.

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

A: To simplify an expression with a negative exponent, you need to take the reciprocal of the base and change the sign of the exponent. For example, 4−24^{-2} can be simplified as 142=116\frac{1}{4^2} = \frac{1}{16}.

Q: What is the rule for simplifying a radical expression with a coefficient?

A: The rule for simplifying a radical expression with a coefficient is to simplify the radical expression first and then multiply the result by the coefficient. For example, 2162\sqrt{16} can be simplified as 2⋅4=82 \cdot 4 = 8.

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

A: To simplify an expression with multiple exponents, you need to evaluate each exponent separately and then combine the results. For example, 42⋅324^2 \cdot 3^2 can be simplified as 16⋅9=14416 \cdot 9 = 144.

Q: What is the difference between a rational and irrational number?

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction. For example, 12\frac{1}{2} is a rational number, while 2\sqrt{2} is an irrational number.

Q: How do I simplify an expression with a rational and irrational number?

A: To simplify an expression with a rational and irrational number, you need to evaluate each number separately and then combine the results. For example, 12+2\frac{1}{2} + \sqrt{2} can be simplified as 12+1.414=1.707\frac{1}{2} + 1.414 = 1.707.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about simplifying expressions. We discussed the rules and properties of exponents and radicals, and provided examples to illustrate each concept. We hope that this article has been helpful in clarifying any confusion you may have had about simplifying expressions.

Final Answer

The final answer is that simplifying expressions is an essential skill in mathematics, and understanding the rules and properties of exponents and radicals is crucial for simplifying expressions.