Simplify $7t - (3 - 25t) + \sqrt{-81}$ And Write In Standard Form.A. $-3 - 9i$ B. $-3 + 41i$ C. $29i$ D. $-3 - 41i$ E. $3 + 41t$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. In this article, we will simplify the given expression $7t - (3 - 25t) + \sqrt{-81}$ and write it in standard form.

Step 1: Simplify the Expression Inside the Parentheses

The first step is to simplify the expression inside the parentheses. We can start by evaluating the expression 3−25t3 - 25t. This expression is already simplified, so we can move on to the next step.

Step 2: Distribute the Negative Sign

The next step is to distribute the negative sign to the expression inside the parentheses. This means that we need to multiply the negative sign by each term inside the parentheses. Using the distributive property, we get:

−(3−25t)=−3+25t- (3 - 25t) = -3 + 25t

Step 3: Simplify the Expression

Now that we have simplified the expression inside the parentheses, we can combine like terms. We can start by combining the terms with the variable tt. We have:

7t+25t=32t7t + 25t = 32t

Step 4: Simplify the Square Root

The next step is to simplify the square root. We have:

−81=(−9)2=9i\sqrt{-81} = \sqrt{(-9)^2} = 9i

Step 5: Combine Like Terms

Now that we have simplified the square root, we can combine like terms. We have:

32t−3+9i32t - 3 + 9i

Step 6: Write in Standard Form

The final step is to write the expression in standard form. We can start by combining the constant terms. We have:

−3+9i-3 + 9i

Conclusion

In this article, we simplified the given expression $7t - (3 - 25t) + \sqrt{-81}$ and wrote it in standard form. We followed the steps involved in simplifying algebraic expressions, including simplifying the expression inside the parentheses, distributing the negative sign, simplifying the expression, simplifying the square root, combining like terms, and writing in standard form.

Answer

The final answer is:

−3+9i-3 + 9i

Discussion

This expression can be written in the form a+bia + bi, where aa and bb are real numbers. In this case, we have a=−3a = -3 and b=9b = 9. This is a complex number, and it can be represented on the complex plane.

Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. They can be represented in the form a+bia + bi, where aa and bb are real numbers. The real part of a complex number is the part that is not multiplied by the imaginary unit ii, while the imaginary part is the part that is multiplied by ii.

Imaginary Unit

The imaginary unit ii is a number that satisfies the equation i2=−1i^2 = -1. It is used to represent the imaginary part of a complex number. When we multiply a complex number by ii, we are essentially rotating the number by 90∘90^\circ counterclockwise in the complex plane.

Complex Plane

The complex plane is a two-dimensional plane that is used to represent complex numbers. It is similar to the Cartesian plane, but it has an additional axis that represents the imaginary part of a complex number. The complex plane is used to visualize complex numbers and their relationships.

Conclusion

Introduction

In our previous article, we simplified the given expression $7t - (3 - 25t) + \sqrt{-81}$ and wrote it in standard form. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to simplify the expression inside the parentheses. This involves evaluating any expressions that are inside the parentheses and simplifying them.

Q: How do I distribute a negative sign to an expression inside parentheses?

A: To distribute a negative sign to an expression inside parentheses, you need to multiply the negative sign by each term inside the parentheses. This is done using the distributive property.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms inside parentheses. It is used to simplify expressions and is a fundamental concept in algebra.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. This is done by factoring the number inside the square root and identifying the largest perfect square factor.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without any imaginary part. An imaginary number, on the other hand, is a number that has an imaginary part. Imaginary numbers are used to represent complex numbers and are denoted by the letter ii.

Q: How do I write a complex number in standard form?

A: To write a complex number in standard form, you need to express it in the form a+bia + bi, where aa and bb are real numbers. The real part of the complex number is the part that is not multiplied by the imaginary unit ii, while the imaginary part is the part that is multiplied by ii.

Q: What is the complex plane?

A: The complex plane is a two-dimensional plane that is used to represent complex numbers. It is similar to the Cartesian plane, but it has an additional axis that represents the imaginary part of a complex number.

Q: How do I visualize complex numbers on the complex plane?

A: To visualize complex numbers on the complex plane, you need to plot the real part of the complex number on the x-axis and the imaginary part on the y-axis. This will give you a graphical representation of the complex number.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not simplifying expressions inside parentheses
  • Not distributing negative signs correctly
  • Not simplifying square roots
  • Not writing complex numbers in standard form
  • Not visualizing complex numbers on the complex plane

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. We answered some frequently asked questions about simplifying algebraic expressions and provided tips and tricks to help you avoid common mistakes.

Additional Resources

For more information on simplifying algebraic expressions, we recommend the following resources:

  • Khan Academy: Algebra
  • Mathway: Algebra
  • Wolfram Alpha: Algebra

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

  • Simplify the expression 2x−(3+4x)+162x - (3 + 4x) + \sqrt{16}
  • Simplify the expression 3y+(2−5y)+93y + (2 - 5y) + \sqrt{9}
  • Simplify the expression 4z−(2+3z)+254z - (2 + 3z) + \sqrt{25}

Answer Key

  1. −x+5-x + 5
  2. −3y+2-3y + 2
  3. −2z+3-2z + 3