Subtract: ${ -15b^7 - \left(-13b^7 + 11\right) }$Your Answer Should Be In Simplest Terms.

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Understanding the Concept of Subtraction in Algebra

When dealing with algebraic expressions, subtraction is an essential operation that helps us simplify complex equations. In this article, we will focus on subtracting one algebraic expression from another, with a specific emphasis on simplifying the resulting expression.

The Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is crucial in mathematics, as it helps us to:

  • Reduce complexity: Simplifying expressions makes them easier to work with, reducing the risk of errors and increasing the efficiency of calculations.
  • Identify patterns: By simplifying expressions, we can identify patterns and relationships between variables, which is essential in solving equations and inequalities.
  • Make calculations easier: Simplified expressions make it easier to perform calculations, as we can work with smaller and more manageable expressions.

Subtracting Algebraic Expressions: A Step-by-Step Approach

To subtract one algebraic expression from another, we need to follow a step-by-step approach:

  1. Distribute the negative sign: When subtracting an expression, we need to distribute the negative sign to each term inside the parentheses.
  2. Combine like terms: After distributing the negative sign, we can combine like terms to simplify the expression.
  3. Simplify the resulting expression: Finally, we need to simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Example: Subtracting (−13b7+11)\left(-13b^7 + 11\right) from −15b7-15b^7

Let's apply the step-by-step approach to subtract (−13b7+11)\left(-13b^7 + 11\right) from −15b7-15b^7:

Step 1: Distribute the Negative Sign

When subtracting (−13b7+11)\left(-13b^7 + 11\right) from −15b7-15b^7, we need to distribute the negative sign to each term inside the parentheses:

−15b7−(−13b7+11)-15b^7 - \left(-13b^7 + 11\right)

Step 2: Combine Like Terms

After distributing the negative sign, we can combine like terms to simplify the expression:

−15b7+13b7−11-15b^7 + 13b^7 - 11

Step 3: Simplify the Resulting Expression

Finally, we need to simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses:

−2b7−11-2b^7 - 11

Conclusion

In conclusion, subtracting algebraic expressions is a crucial operation in mathematics that helps us simplify complex equations. By following a step-by-step approach, we can distribute the negative sign, combine like terms, and simplify the resulting expression. In this article, we applied this approach to subtract (−13b7+11)\left(-13b^7 + 11\right) from −15b7-15b^7, resulting in the simplified expression −2b7−11-2b^7 - 11.

Frequently Asked Questions

Q: What is the difference between subtracting and distributing in algebra?

A: Subtracting and distributing are two different operations in algebra. Subtracting involves removing a term or expression from another, while distributing involves multiplying a term or expression by each term inside parentheses.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms, eliminate any unnecessary parentheses, and reduce the expression to its simplest form.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is crucial in mathematics, as it helps us reduce complexity, identify patterns, and make calculations easier.

Further Reading

For more information on simplifying algebraic expressions, check out the following resources:

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Q: What is the difference between subtracting and distributing in algebra?

A: Subtracting involves removing a term or expression from another, while distributing involves multiplying a term or expression by each term inside parentheses.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to:

  • Combine like terms: Combine terms that have the same variable and exponent.
  • Eliminate unnecessary parentheses: Remove any unnecessary parentheses to make the expression easier to read.
  • Reduce the expression to its simplest form: Simplify the expression as much as possible.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is crucial in mathematics because it:

  • Reduces complexity: Simplifying expressions makes them easier to work with, reducing the risk of errors and increasing the efficiency of calculations.
  • Identifies patterns: By simplifying expressions, we can identify patterns and relationships between variables, which is essential in solving equations and inequalities.
  • Makes calculations easier: Simplified expressions make it easier to perform calculations, as we can work with smaller and more manageable expressions.

Q: How do I handle negative signs when simplifying expressions?

A: When simplifying expressions, you need to handle negative signs carefully. Here are some tips:

  • Distribute the negative sign: When subtracting an expression, distribute the negative sign to each term inside the parentheses.
  • Combine like terms: Combine terms that have the same variable and exponent, including terms with negative signs.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the difference between a variable and a constant in algebra?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I handle exponents when simplifying expressions?

A: When simplifying expressions, you need to handle exponents carefully. Here are some tips:

  • Apply the exponent rules: Apply the exponent rules, such as multiplying exponents when multiplying terms with the same base.
  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the importance of using parentheses when simplifying expressions?

A: Using parentheses is essential when simplifying expressions because it:

  • Clarifies the order of operations: Parentheses help to clarify the order of operations, ensuring that we perform calculations in the correct order.
  • Prevents errors: Using parentheses helps to prevent errors by making it clear which terms are being added or subtracted.
  • Makes calculations easier: Using parentheses makes it easier to perform calculations, as we can work with smaller and more manageable expressions.

Q: How do I simplify expressions with multiple variables?

A: When simplifying expressions with multiple variables, you need to:

  • Combine like terms: Combine terms that have the same variable and exponent.
  • Apply the exponent rules: Apply the exponent rules, such as multiplying exponents when multiplying terms with the same base.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression with a single variable and a degree of 1, while a quadratic expression is an expression with a single variable and a degree of 2.

Q: How do I simplify expressions with fractions?

A: When simplifying expressions with fractions, you need to:

  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify the fraction: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the importance of using a calculator when simplifying expressions?

A: Using a calculator can be helpful when simplifying expressions, especially when dealing with complex calculations or large numbers. However, it's essential to remember that a calculator is only a tool, and you should always double-check your work to ensure accuracy.

Q: How do I check my work when simplifying expressions?

A: When checking your work, you should:

  • Re-read the problem: Re-read the problem to ensure you understand what is being asked.
  • Re-check your calculations: Re-check your calculations to ensure you have performed the correct operations.
  • Simplify the expression: Simplify the expression to ensure it is in its simplest form.

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

A: A rational expression is an expression that can be written as a fraction, while an irrational expression is an expression that cannot be written as a fraction.

Q: How do I simplify expressions with radicals?

A: When simplifying expressions with radicals, you need to:

  • Simplify the radical: Simplify the radical by finding the square root of the number inside the radical.
  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the importance of using a graphing calculator when simplifying expressions?

A: Using a graphing calculator can be helpful when simplifying expressions, especially when dealing with complex calculations or large numbers. However, it's essential to remember that a graphing calculator is only a tool, and you should always double-check your work to ensure accuracy.

Q: How do I check my work when simplifying expressions with a graphing calculator?

A: When checking your work with a graphing calculator, you should:

  • Re-read the problem: Re-read the problem to ensure you understand what is being asked.
  • Re-check your calculations: Re-check your calculations to ensure you have performed the correct operations.
  • Simplify the expression: Simplify the expression to ensure it is in its simplest form.

Q: What is the difference between a polynomial expression and a non-polynomial expression?

A: A polynomial expression is an expression that consists of variables and constants combined using only addition, subtraction, and multiplication, while a non-polynomial expression is an expression that does not meet these criteria.

Q: How do I simplify expressions with absolute value?

A: When simplifying expressions with absolute value, you need to:

  • Simplify the absolute value: Simplify the absolute value by removing the absolute value symbol and considering both positive and negative cases.
  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the importance of using a computer algebra system (CAS) when simplifying expressions?

A: Using a CAS can be helpful when simplifying expressions, especially when dealing with complex calculations or large numbers. However, it's essential to remember that a CAS is only a tool, and you should always double-check your work to ensure accuracy.

Q: How do I check my work when simplifying expressions with a CAS?

A: When checking your work with a CAS, you should:

  • Re-read the problem: Re-read the problem to ensure you understand what is being asked.
  • Re-check your calculations: Re-check your calculations to ensure you have performed the correct operations.
  • Simplify the expression: Simplify the expression to ensure it is in its simplest form.

Q: What is the difference between a numerical expression and a symbolic expression?

A: A numerical expression is an expression that consists of numbers and operations, while a symbolic expression is an expression that consists of variables and operations.

Q: How do I simplify expressions with trigonometric functions?

A: When simplifying expressions with trigonometric functions, you need to:

  • Simplify the trigonometric function: Simplify the trigonometric function by using trigonometric identities and formulas.
  • Combine like terms: Combine terms that have the same variable and exponent.
  • Simplify the resulting expression: Simplify the resulting expression by combining like terms and eliminating any unnecessary parentheses.

Q: What is the importance of using a graphing calculator when simplifying expressions with trigonometric functions?

A: Using a graphing calculator can be helpful when simplifying expressions with trigonometric functions, especially when dealing with complex calculations or large numbers. However, it's essential to remember that a graphing calculator is only a tool, and you should always double-check your work to ensure accuracy.

Q: How do I check my work when simplifying expressions with trigonometric functions and a graphing calculator?

A: When checking your work with a graphing calculator, you should:

  • Re-read the problem: Re-read the problem to ensure you understand what is being asked.
  • Re-check your calculations: Re-check your calculations to ensure you have performed the correct operations.
  • Simplify the expression: Simplify the expression to ensure it is in its simplest form.

Q: What is the difference between a rational function and