Evaluate: ${ -p^0 - P^2(p - A^0) - Ap + |-ap| }$if { A = -3 $}$ And { P = -4 $}$.

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Introduction

When evaluating mathematical expressions, it's essential to follow the order of operations (PEMDAS/BODMAS) to ensure accuracy. In this article, we will evaluate the given expression ${ -p^0 - p^2(p - a^0) - ap + |-ap| }$ when a=−3a = -3 and p=−4p = -4. We will break down the expression step by step, applying the rules of arithmetic operations and absolute value.

Understanding the Expression

The given expression is ${ -p^0 - p^2(p - a^0) - ap + |-ap| }$. Let's analyze each part of the expression:

  • −p0-p^0: This term represents the negation of pp raised to the power of 0. Since any number raised to the power of 0 is 1, −p0-p^0 simplifies to −1-1.
  • p2(p−a0)p^2(p - a^0): This term involves squaring pp and then multiplying it by the result of subtracting a0a^0 from pp. As mentioned earlier, a0a^0 is 1, so this term simplifies to p3−pp^3 - p.
  • −ap-ap: This term represents the negation of the product of aa and pp.
  • ∣−ap∣|-ap|: This term represents the absolute value of the product of aa and pp.

Substituting Values

We are given that a=−3a = -3 and p=−4p = -4. Let's substitute these values into the expression:

  • −p0-p^0 becomes −(−4)0=−1-(-4)^0 = -1
  • p2(p−a0)p^2(p - a^0) becomes (−4)3−(−4)=−64+4=−60(-4)^3 - (-4) = -64 + 4 = -60
  • −ap-ap becomes −(−3)(−4)=12-(-3)(-4) = 12
  • ∣−ap∣|-ap| becomes ∣−(−3)(−4)∣=12|-(-3)(-4)| = 12

Evaluating the Expression

Now that we have substituted the values, let's evaluate the expression:

{ -p^0 - p^2(p - a^0) - ap + |-ap| \}

=−1−(−60)−12+12= -1 - (-60) - 12 + 12

=−1+60−12+12= -1 + 60 - 12 + 12

=59= 59

Conclusion

In this article, we evaluated the given expression ${ -p^0 - p^2(p - a^0) - ap + |-ap| }$ when a=−3a = -3 and p=−4p = -4. By following the order of operations and applying the rules of arithmetic operations and absolute value, we simplified the expression and obtained a final value of 59.

Frequently Asked Questions

  • What is the order of operations (PEMDAS/BODMAS)?
    • The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.
  • What is the rule for absolute value?
    • The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, it is the non-negative value of a number.
  • How do you simplify expressions involving absolute value?
    • To simplify expressions involving absolute value, you can remove the absolute value sign and consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. Then, you can simplify each case separately and combine the results.

Further Reading

References

Introduction

Evaluating mathematical expressions can be a challenging task, especially when dealing with complex expressions involving absolute value, exponents, and multiple operations. In this article, we will address some common questions and provide step-by-step solutions to help you evaluate mathematical expressions with confidence.

Q&A

Q1: What is the order of operations (PEMDAS/BODMAS)?

A1: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

Q2: How do I simplify expressions involving absolute value?

A2: To simplify expressions involving absolute value, you can remove the absolute value sign and consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. Then, you can simplify each case separately and combine the results.

Q3: What is the rule for evaluating expressions with exponents?

A3: When evaluating expressions with exponents, you should follow the order of operations (PEMDAS/BODMAS). First, evaluate any expressions inside parentheses or brackets. Then, evaluate any exponents (such as squaring or cubing). Finally, perform any multiplication and division operations from left to right.

Q4: How do I handle negative exponents?

A4: When dealing with negative exponents, you can rewrite the expression as a fraction. For example, a−n=1ana^{-n} = \frac{1}{a^n}. This allows you to simplify the expression and perform the necessary operations.

Q5: What is the difference between a variable and a constant?

A5: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same. In mathematical expressions, variables are often represented by letters such as x, y, or z, while constants are represented by numbers.

Q6: How do I evaluate expressions with multiple operations?

A6: When evaluating expressions with multiple operations, you should follow the order of operations (PEMDAS/BODMAS). First, evaluate any expressions inside parentheses or brackets. Then, evaluate any exponents (such as squaring or cubing). Finally, perform any multiplication and division operations from left to right, and then perform any addition and subtraction operations from left to right.

Q7: What is the rule for evaluating expressions with absolute value and exponents?

A7: When evaluating expressions with absolute value and exponents, you should follow the order of operations (PEMDAS/BODMAS). First, evaluate any expressions inside parentheses or brackets. Then, evaluate any exponents (such as squaring or cubing). Next, evaluate the absolute value expression. Finally, perform any multiplication and division operations from left to right, and then perform any addition and subtraction operations from left to right.

Example Solutions

Example 1: Evaluating an Expression with Absolute Value

Evaluate the expression ${ |-3x + 4| }$ when x=2x = 2.

Solution:

  • First, substitute x=2x = 2 into the expression: ${ |-3(2) + 4| }$
  • Next, evaluate the expression inside the absolute value: ${ |-6 + 4| }$
  • Then, simplify the expression: ${ |-2| }$
  • Finally, evaluate the absolute value: ${ 2 }$

Example 2: Evaluating an Expression with Exponents

Evaluate the expression ${ 2^3 + 3^2 }$.

Solution:

  • First, evaluate the exponents: ${ 8 + 9 }$
  • Next, simplify the expression: ${ 17 }$

Example 3: Evaluating an Expression with Multiple Operations

Evaluate the expression ${ 2x + 3y - 4 }$ when x=2x = 2 and y=3y = 3.

Solution:

  • First, substitute x=2x = 2 and y=3y = 3 into the expression: ${ 2(2) + 3(3) - 4 }$
  • Next, evaluate the expression: ${ 4 + 9 - 4 }$
  • Then, simplify the expression: ${ 9 }$

Conclusion

Evaluating mathematical expressions can be a challenging task, but by following the order of operations (PEMDAS/BODMAS) and understanding the rules for absolute value, exponents, and multiple operations, you can simplify complex expressions and arrive at accurate solutions. Remember to always follow the order of operations and to simplify expressions step by step.

Frequently Asked Questions

  • What is the order of operations (PEMDAS/BODMAS)?
    • The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.
  • How do I simplify expressions involving absolute value?
    • To simplify expressions involving absolute value, you can remove the absolute value sign and consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. Then, you can simplify each case separately and combine the results.
  • What is the rule for evaluating expressions with exponents?
    • When evaluating expressions with exponents, you should follow the order of operations (PEMDAS/BODMAS). First, evaluate any expressions inside parentheses or brackets. Then, evaluate any exponents (such as squaring or cubing). Finally, perform any multiplication and division operations from left to right.

Further Reading

References