Applying Exponent Rules to the Given Equation

Oct 13, 2025 by miodeya.com 66 views

Solving for the Values of a and b in the Given Equation

In this article, we will explore the concept of solving for the values of a and b in a given equation involving exponents. The equation provided is $\left(5 x^7 y^2\right)\left(-4 x^4 y^5\right)=-20 x^a y^b$ . Our goal is to determine the values of a and b that make this equation true.

Before we dive into solving the equation, it's essential to understand the rules of exponents. When multiplying two numbers with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$ . Similarly, when dividing two numbers with the same base, we subtract their exponents. For instance, $\frac{x^a}{x^b} = x^{a-b}$ .

Applying Exponent Rules to the Given Equation

Now that we have a solid understanding of exponent rules, let's apply them to the given equation. We can start by multiplying the two expressions on the left-hand side of the equation:

$\left(5 x^7 y^2\right)\left(-4 x^4 y^5\right) = 5 \cdot -4 \cdot x^{7+4} \cdot y^{2+5}$

Using the exponent rules, we can simplify the expression:

$-20 x^{11} y^7$

Now, we can equate this expression to the right-hand side of the equation:

$-20 x^{11} y^7 = -20 x^a y^b$

To solve for a and b, we can compare the exponents of x and y on both sides of the equation. We can see that the exponent of x on the left-hand side is 11, and the exponent of y is 7. Therefore, we can conclude that:

$a = 11$ $b = 7$

In this article, we have solved for the values of a and b in the given equation. By applying exponent rules and comparing the exponents of x and y on both sides of the equation, we have determined that:

$a = 11$ $b = 7$

This solution is consistent with option A, which states that $a = 11, b = 7$ .

The final answer is:

A. $a = 11, b = 7$
Frequently Asked Questions (FAQs) About Solving for a and b in the Given Equation

In our previous article, we explored the concept of solving for the values of a and b in a given equation involving exponents. We applied exponent rules and compared the exponents of x and y on both sides of the equation to determine the values of a and b. In this article, we will address some frequently asked questions (FAQs) about solving for a and b in the given equation.

Q: What is the significance of exponent rules in solving for a and b?

A: Exponent rules are crucial in solving for a and b because they allow us to simplify the expression on the left-hand side of the equation. By applying exponent rules, we can combine like terms and compare the exponents of x and y on both sides of the equation.

Q: How do I know which exponent rule to apply in a given situation?

A: To determine which exponent rule to apply, you need to analyze the expression and identify the base and the exponents. If the bases are the same, you can add or subtract the exponents depending on whether you are multiplying or dividing. If the bases are different, you cannot combine the exponents.

Q: What if the equation has multiple variables with different bases? How do I solve for a and b?

A: If the equation has multiple variables with different bases, you need to apply the exponent rules separately for each variable. For example, if the equation is $\left(5 x^7 y^2\right)\left(-4 x^4 z^3\right)=-20 x^a y^b z^c$ , you can apply the exponent rules separately for x, y, and z.

Q: Can I use the same exponent rule for all variables in the equation?

A: No, you cannot use the same exponent rule for all variables in the equation. Each variable has its own set of exponent rules, and you need to apply them separately.

Q: What if the equation has a negative exponent? How do I handle it?

A: If the equation has a negative exponent, you can handle it by applying the rule for negative exponents. For example, if the equation is $x^{-a} = \frac{1}{x^a}$ , you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base.

Q: Can I use a calculator to solve for a and b?

A: While a calculator can be a useful tool in solving for a and b, it is not always necessary. In many cases, you can solve for a and b by applying exponent rules and comparing the exponents of x and y on both sides of the equation.

Q: What if I get stuck while solving for a and b? Where can I find help?

A: If you get stuck while solving for a and b, you can try the following:

Review the exponent rules and make sure you understand them.
Break down the equation into smaller parts and solve each part separately.
Use online resources, such as video tutorials or practice problems, to help you understand the concept.
Ask a teacher or tutor for help.

In this article, we have addressed some frequently asked questions (FAQs) about solving for a and b in the given equation. We have provided explanations and examples to help you understand the concept and apply it to different situations. By following the tips and advice provided in this article, you can become more confident and proficient in solving for a and b in equations involving exponents.