Simplify The Expression: 256 ( 1 2 − 1 4 ) 1 2 \sqrt[\frac{1}{2}]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}} 2 1 ​ 25 6 ( 2 1 ​ − 4 1 ​ ) ​

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Introduction

Mathematical expressions can be complex and challenging to simplify, especially when they involve exponents, roots, and fractions. In this article, we will focus on simplifying the given expression: \sqrt[\frac{1]{2}]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}}. We will break down the expression into manageable parts, apply the rules of exponents and roots, and finally arrive at the simplified form.

Understanding the Expression

The given expression involves a square root with a fractional exponent. To simplify it, we need to understand the properties of exponents and roots. The expression can be rewritten as:

256(12−14)12=256(12−14)2\sqrt[\frac{1}{2}]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}} = \sqrt[2]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}}

Evaluating the Exponent

The first step is to evaluate the exponent (12−14)\left(\frac{1}{2}-\frac{1}{4}\right). To do this, we need to find a common denominator, which is 4.

(12−14)=(24−14)=14\left(\frac{1}{2}-\frac{1}{4}\right) = \left(\frac{2}{4}-\frac{1}{4}\right) = \frac{1}{4}

Simplifying the Expression

Now that we have evaluated the exponent, we can simplify the expression:

256(12−14)2=256142\sqrt[2]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}} = \sqrt[2]{256^{\frac{1}{4}}}

Applying the Power of a Power Rule

The next step is to apply the power of a power rule, which states that (am)n=amn\left(a^m\right)^n = a^{mn}. In this case, we have:

256142=(25612)122\sqrt[2]{256^{\frac{1}{4}}} = \sqrt[2]{\left(256^{\frac{1}{2}}\right)^{\frac{1}{2}}}

Simplifying the Expression Further

Now we can simplify the expression further by evaluating the exponent 12\frac{1}{2}:

(25612)122=(16)122\sqrt[2]{\left(256^{\frac{1}{2}}\right)^{\frac{1}{2}}} = \sqrt[2]{\left(16\right)^{\frac{1}{2}}}

Evaluating the Square Root

The final step is to evaluate the square root:

(16)122=42\sqrt[2]{\left(16\right)^{\frac{1}{2}}} = \sqrt[2]{4}

Conclusion

In conclusion, we have successfully simplified the given expression: 256(12−14)12\sqrt[\frac{1}{2}]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}}. By breaking down the expression into manageable parts, applying the rules of exponents and roots, and finally arriving at the simplified form, we have demonstrated the importance of understanding the properties of mathematical expressions.

Frequently Asked Questions

  • What is the simplified form of the given expression?
  • How do we evaluate the exponent in the given expression?
  • What is the power of a power rule, and how do we apply it in this case?

Final Answer

The final answer is 2\boxed{2}.

Step-by-Step Solution

  1. Evaluate the exponent (12−14)\left(\frac{1}{2}-\frac{1}{4}\right).
  2. Simplify the expression by applying the power of a power rule.
  3. Evaluate the square root of the resulting expression.
  4. The final answer is 2\boxed{2}.

Additional Resources

For more information on simplifying mathematical expressions, please refer to the following resources:

  • Khan Academy: Simplifying Exponents and Roots
  • Mathway: Simplifying Expressions with Exponents and Roots
  • Wolfram Alpha: Simplifying Mathematical Expressions

Conclusion

In this article, we have demonstrated the importance of understanding the properties of mathematical expressions. By breaking down the given expression into manageable parts, applying the rules of exponents and roots, and finally arriving at the simplified form, we have shown that simplifying mathematical expressions is a step-by-step process that requires patience and attention to detail.

Introduction

In our previous article, we simplified the expression: 256(12−14)12\sqrt[\frac{1}{2}]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}}. We broke down the expression into manageable parts, applied the rules of exponents and roots, and finally arrived at the simplified form. In this article, we will answer some frequently asked questions related to the simplification of the given expression.

Q&A

Q: What is the simplified form of the given expression?

A: The simplified form of the given expression is 2\boxed{2}.

Q: How do we evaluate the exponent in the given expression?

A: To evaluate the exponent, we need to find a common denominator, which is 4. Then, we can subtract the fractions: (12−14)=(24−14)=14\left(\frac{1}{2}-\frac{1}{4}\right) = \left(\frac{2}{4}-\frac{1}{4}\right) = \frac{1}{4}.

Q: What is the power of a power rule, and how do we apply it in this case?

A: The power of a power rule states that (am)n=amn\left(a^m\right)^n = a^{mn}. In this case, we have: 256142=(25612)122\sqrt[2]{256^{\frac{1}{4}}} = \sqrt[2]{\left(256^{\frac{1}{2}}\right)^{\frac{1}{2}}}.

Q: How do we simplify the expression further?

A: We can simplify the expression further by evaluating the exponent 12\frac{1}{2}: (25612)122=(16)122\sqrt[2]{\left(256^{\frac{1}{2}}\right)^{\frac{1}{2}}} = \sqrt[2]{\left(16\right)^{\frac{1}{2}}}.

Q: What is the final step in simplifying the expression?

A: The final step is to evaluate the square root: (16)122=42\sqrt[2]{\left(16\right)^{\frac{1}{2}}} = \sqrt[2]{4}.

Q: What is the importance of understanding the properties of mathematical expressions?

A: Understanding the properties of mathematical expressions is crucial in simplifying complex expressions. By breaking down the expression into manageable parts, applying the rules of exponents and roots, and finally arriving at the simplified form, we can demonstrate the importance of understanding the properties of mathematical expressions.

Q: What are some additional resources for learning about simplifying mathematical expressions?

A: Some additional resources for learning about simplifying mathematical expressions include:

  • Khan Academy: Simplifying Exponents and Roots
  • Mathway: Simplifying Expressions with Exponents and Roots
  • Wolfram Alpha: Simplifying Mathematical Expressions

Conclusion

In this article, we have answered some frequently asked questions related to the simplification of the given expression: 256(12−14)12\sqrt[\frac{1}{2}]{256^{\left(\frac{1}{2}-\frac{1}{4}\right)}}. We have demonstrated the importance of understanding the properties of mathematical expressions and provided some additional resources for learning about simplifying mathematical expressions.

Frequently Asked Questions

  • What is the simplified form of the given expression?
  • How do we evaluate the exponent in the given expression?
  • What is the power of a power rule, and how do we apply it in this case?
  • How do we simplify the expression further?
  • What is the final step in simplifying the expression?
  • What is the importance of understanding the properties of mathematical expressions?
  • What are some additional resources for learning about simplifying mathematical expressions?

Final Answer

The final answer is 2\boxed{2}.

Step-by-Step Solution

  1. Evaluate the exponent (12−14)\left(\frac{1}{2}-\frac{1}{4}\right).
  2. Simplify the expression by applying the power of a power rule.
  3. Evaluate the square root of the resulting expression.
  4. The final answer is 2\boxed{2}.

Additional Resources

For more information on simplifying mathematical expressions, please refer to the following resources:

  • Khan Academy: Simplifying Exponents and Roots
  • Mathway: Simplifying Expressions with Exponents and Roots
  • Wolfram Alpha: Simplifying Mathematical Expressions