Which Values Of $a$ And $b$ Make The Following Equation True?$\[ (5x^7 Y^2)(-4x^4 Y^5) = -20x^a Y^b \\]A. $a = 11$, $b = 7$B. $a = 11$, $b = 10$C. $a = 28$, $b =

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Solving for a and b in the Given Equation

In this article, we will explore the values of aa and bb that make the equation (5x7y2)(βˆ’4x4y5)=βˆ’20xayb(5x^7 y^2)(-4x^4 y^5) = -20x^a y^b true. This equation involves the multiplication of two polynomials, resulting in a new polynomial with unknown exponents. Our goal is to determine the values of aa and bb that satisfy this equation.

Understanding Exponents and Multiplication

Before we dive into solving the equation, let's review the rules of exponents and multiplication. When multiplying two variables with the same base, we add their exponents. For example, xaβ‹…xb=xa+bx^a \cdot x^b = x^{a+b}. Similarly, when multiplying two variables with different bases, we multiply their coefficients and add their exponents. For example, xaβ‹…yb=xaybx^a \cdot y^b = x^a y^b.

Applying the Rules of Exponents and Multiplication

Now, let's apply these rules to the given equation. We have (5x7y2)(βˆ’4x4y5)=βˆ’20xayb(5x^7 y^2)(-4x^4 y^5) = -20x^a y^b. Using the rules of exponents and multiplication, we can simplify the left-hand side of the equation as follows:

(5x7y2)(βˆ’4x4y5)=βˆ’20x7+4y2+5(5x^7 y^2)(-4x^4 y^5) = -20x^{7+4} y^{2+5} =βˆ’20x11y7= -20x^{11} y^7

Comparing the Left-Hand Side and Right-Hand Side

Now, let's compare the left-hand side and right-hand side of the equation. We have βˆ’20x11y7=βˆ’20xayb-20x^{11} y^7 = -20x^a y^b. Since the coefficients are the same, we can focus on the exponents. We have 11=a11 = a and 7=b7 = b.

In conclusion, the values of aa and bb that make the equation true are a=11a = 11 and b=7b = 7. This is the only solution that satisfies the equation.

The correct answer is:

  • A. a=11a = 11, b=7b = 7

This problem involves the application of the rules of exponents and multiplication. It requires the student to simplify the left-hand side of the equation and compare it with the right-hand side. The student must also be able to identify the values of aa and bb that satisfy the equation.

  • Make sure to apply the rules of exponents and multiplication correctly.
  • Simplify the left-hand side of the equation before comparing it with the right-hand side.
  • Identify the values of aa and bb that satisfy the equation.
  • Solve the equation (3x2y3)(βˆ’2x4y2)=βˆ’6xayb(3x^2 y^3)(-2x^4 y^2) = -6x^a y^b.
  • Solve the equation (4x5y4)(βˆ’3x2y3)=βˆ’12xayb(4x^5 y^4)(-3x^2 y^3) = -12x^a y^b.

In this article, we have solved the equation (5x7y2)(βˆ’4x4y5)=βˆ’20xayb(5x^7 y^2)(-4x^4 y^5) = -20x^a y^b and determined the values of aa and bb that satisfy the equation. We have also provided tips and tricks for solving similar problems.
Q&A: Solving for a and b in the Given Equation

In our previous article, we solved the equation (5x7y2)(βˆ’4x4y5)=βˆ’20xayb(5x^7 y^2)(-4x^4 y^5) = -20x^a y^b and determined the values of aa and bb that satisfy the equation. In this article, we will answer some frequently asked questions related to this topic.

Q: What are the rules of exponents and multiplication?

A: The rules of exponents and multiplication are as follows:

  • When multiplying two variables with the same base, we add their exponents. For example, xaβ‹…xb=xa+bx^a \cdot x^b = x^{a+b}.
  • When multiplying two variables with different bases, we multiply their coefficients and add their exponents. For example, xaβ‹…yb=xaybx^a \cdot y^b = x^a y^b.

Q: How do I simplify the left-hand side of the equation?

A: To simplify the left-hand side of the equation, you need to apply the rules of exponents and multiplication. For example, in the equation (5x7y2)(βˆ’4x4y5)=βˆ’20xayb(5x^7 y^2)(-4x^4 y^5) = -20x^a y^b, you can simplify the left-hand side as follows:

(5x7y2)(βˆ’4x4y5)=βˆ’20x7+4y2+5(5x^7 y^2)(-4x^4 y^5) = -20x^{7+4} y^{2+5} =βˆ’20x11y7= -20x^{11} y^7

Q: How do I compare the left-hand side and right-hand side of the equation?

A: To compare the left-hand side and right-hand side of the equation, you need to identify the values of aa and bb that satisfy the equation. In the equation (5x7y2)(βˆ’4x4y5)=βˆ’20xayb(5x^7 y^2)(-4x^4 y^5) = -20x^a y^b, you can compare the left-hand side and right-hand side as follows:

βˆ’20x11y7=βˆ’20xayb-20x^{11} y^7 = -20x^a y^b

Since the coefficients are the same, you can focus on the exponents. You have 11=a11 = a and 7=b7 = b.

Q: What are some tips and tricks for solving similar problems?

A: Here are some tips and tricks for solving similar problems:

  • Make sure to apply the rules of exponents and multiplication correctly.
  • Simplify the left-hand side of the equation before comparing it with the right-hand side.
  • Identify the values of aa and bb that satisfy the equation.

Q: Can you provide some practice problems for me to try?

A: Yes, here are some practice problems for you to try:

  • Solve the equation (3x2y3)(βˆ’2x4y2)=βˆ’6xayb(3x^2 y^3)(-2x^4 y^2) = -6x^a y^b.
  • Solve the equation (4x5y4)(βˆ’3x2y3)=βˆ’12xayb(4x^5 y^4)(-3x^2 y^3) = -12x^a y^b.

In this article, we have answered some frequently asked questions related to solving for aa and bb in the given equation. We have also provided some tips and tricks for solving similar problems and some practice problems for you to try.

  • Q: What are the rules of exponents and multiplication?
  • A: The rules of exponents and multiplication are as follows:
    • When multiplying two variables with the same base, we add their exponents. For example, xaβ‹…xb=xa+bx^a \cdot x^b = x^{a+b}.
    • When multiplying two variables with different bases, we multiply their coefficients and add their exponents. For example, xaβ‹…yb=xaybx^a \cdot y^b = x^a y^b.
  • Q: How do I simplify the left-hand side of the equation?
  • A: To simplify the left-hand side of the equation, you need to apply the rules of exponents and multiplication.
  • Q: How do I compare the left-hand side and right-hand side of the equation?
  • A: To compare the left-hand side and right-hand side of the equation, you need to identify the values of aa and bb that satisfy the equation.
  • Q: What are some tips and tricks for solving similar problems?
  • A: Here are some tips and tricks for solving similar problems:
    • Make sure to apply the rules of exponents and multiplication correctly.
    • Simplify the left-hand side of the equation before comparing it with the right-hand side.
    • Identify the values of aa and bb that satisfy the equation.
  • Solve the equation (3x2y3)(βˆ’2x4y2)=βˆ’6xayb(3x^2 y^3)(-2x^4 y^2) = -6x^a y^b.
  • Solve the equation (4x5y4)(βˆ’3x2y3)=βˆ’12xayb(4x^5 y^4)(-3x^2 y^3) = -12x^a y^b.