Evaluate The Expression:${ \left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 }$

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Introduction

In mathematics, expressions are used to represent a value or a relationship between values. Evaluating an expression involves simplifying it to a single value or a simpler form. In this article, we will evaluate the given expression: (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2. We will break down the expression, analyze its components, and provide a step-by-step solution to simplify it.

Understanding the Expression

The given expression consists of three parts:

  1. (b2)2\left(\frac{b}{2}\right)^2
  2. (102)2\left(\frac{10}{2}\right)^2
  3. (−5)2(-5)^2

Each part represents a squared value. To evaluate the expression, we need to simplify each part and then compare the results.

Simplifying the First Part

The first part of the expression is (b2)2\left(\frac{b}{2}\right)^2. To simplify this, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: b2\frac{b}{2}
  2. Square the result: (b2)2=b24\left(\frac{b}{2}\right)^2 = \frac{b^2}{4}

Simplifying the Second Part

The second part of the expression is (102)2\left(\frac{10}{2}\right)^2. To simplify this, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 102=5\frac{10}{2} = 5
  2. Square the result: (102)2=52=25\left(\frac{10}{2}\right)^2 = 5^2 = 25

Simplifying the Third Part

The third part of the expression is (−5)2(-5)^2. To simplify this, we need to follow the order of operations (PEMDAS):

  1. Square the result: (−5)2=25(-5)^2 = 25

Comparing the Results

Now that we have simplified each part of the expression, we can compare the results:

  1. b24\frac{b^2}{4}
  2. 2525
  3. 2525

As we can see, the second and third parts of the expression simplify to the same value: 2525. However, the first part of the expression, b24\frac{b^2}{4}, is not equal to 2525 unless b2=100b^2 = 100.

Conclusion

In conclusion, the given expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 can be simplified to two different values: 2525 and b24\frac{b^2}{4}. The value of bb is not specified in the expression, so we cannot determine the exact value of b24\frac{b^2}{4}. However, we can conclude that the expression is true when b2=100b^2 = 100.

Final Thoughts

Evaluating mathematical expressions is an essential skill in mathematics. By following the order of operations and simplifying expressions, we can gain a deeper understanding of mathematical concepts and relationships. In this article, we evaluated the given expression and simplified it to two different values. We also discussed the importance of understanding the components of an expression and following the order of operations to simplify it.

Frequently Asked Questions

  • Q: What is the value of b24\frac{b^2}{4} when b2=100b^2 = 100? A: When b2=100b^2 = 100, the value of b24\frac{b^2}{4} is 2525.
  • Q: Can the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 be true when b≠10b \neq 10? A: No, the expression cannot be true when b≠10b \neq 10 unless b2=100b^2 = 100.

References

Related Articles

  • Evaluating Expressions with Exponents
  • Simplifying Algebraic Expressions
  • Understanding the Order of Operations

Introduction

In our previous article, we evaluated the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 and simplified it to two different values: 2525 and b24\frac{b^2}{4}. We also discussed the importance of understanding the components of an expression and following the order of operations to simplify it. In this article, we will answer some frequently asked questions related to the expression and provide additional insights.

Q&A

Q: What is the value of b24\frac{b^2}{4} when b2=100b^2 = 100?

A: When b2=100b^2 = 100, the value of b24\frac{b^2}{4} is 2525. This is because 1004=25\frac{100}{4} = 25.

Q: Can the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 be true when b≠10b \neq 10?

A: No, the expression cannot be true when b≠10b \neq 10 unless b2=100b^2 = 100. This is because the expression is only true when the squared values are equal, and b2=100b^2 = 100 is the only value that satisfies this condition.

Q: What is the relationship between the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 and the concept of equality?

A: The expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 illustrates the concept of equality in mathematics. When two expressions are equal, they have the same value or represent the same relationship. In this case, the expression is true when the squared values are equal, and the value of bb is not specified.

Q: How does the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 relate to the concept of variables in mathematics?

A: The expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 involves a variable bb, which represents an unknown value. The expression is true when the value of bb is not specified, and the squared values are equal. This illustrates the concept of variables in mathematics, where a variable can represent a specific value or a range of values.

Q: Can the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 be used to solve for the value of bb?

A: Yes, the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 can be used to solve for the value of bb. When the squared values are equal, we can set up an equation to solve for bb. In this case, we can set up the equation b24=25\frac{b^2}{4} = 25 and solve for bb.

Conclusion

In conclusion, the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 is a useful tool for illustrating the concept of equality and variables in mathematics. By understanding the components of the expression and following the order of operations, we can simplify it and gain insights into the relationship between the variables. We hope that this article has provided additional insights and answered some frequently asked questions related to the expression.

Final Thoughts

Evaluating mathematical expressions is an essential skill in mathematics. By following the order of operations and simplifying expressions, we can gain a deeper understanding of mathematical concepts and relationships. In this article, we evaluated the expression (b2)2=(102)2=(−5)2\left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = (-5)^2 and answered some frequently asked questions related to it. We hope that this article has provided a useful resource for students and educators alike.

Related Articles

  • Evaluating Expressions with Exponents
  • Simplifying Algebraic Expressions
  • Understanding the Order of Operations

References