Fill In The Blanks To Make The Equation True:${ \frac{\left(\square X^8 Y^2 Z^{-3}\right)\left(3 X^4 Y^1 Z^8\right)}{2 X^7 Y^{\square} Z^{\square}}=12 X^{\square} Y^9 }$

Introduction

In this article, we will delve into the world of exponential equations and explore a specific problem that requires us to fill in the blanks to make the equation true. The given equation involves variables with exponents, and our goal is to determine the values of the missing exponents to satisfy the equation. We will break down the problem step by step, using mathematical concepts and techniques to arrive at the solution.

Understanding Exponential Equations

Exponential equations involve variables with exponents, which can be added, subtracted, multiplied, or divided. In this problem, we have an equation with three variables: x, y, and z, each with a specific exponent. Our task is to determine the values of the missing exponents to make the equation true.

The Given Equation

The given equation is:

$$\frac{\left(\square x^8 y^2 z^{-3}\right)\left(3 x^4 y^1 z^8\right)}{2 x^7 y^{\square} z^{\square}}=12 x^{\square} y^9$$

Step 1: Simplify the Left-Hand Side

To simplify the left-hand side of the equation, we can start by multiplying the two expressions inside the parentheses:

$$\left(\square x^8 y^2 z^{-3}\right)\left(3 x^4 y^1 z^8\right) = \square \cdot 3 \cdot x^{8+4} \cdot y^{2+1} \cdot z^{-3+8}$$

Using the laws of exponents, we can combine the exponents:

$$\square \cdot 3 \cdot x^{12} \cdot y^{3} \cdot z^{5}$$

Step 2: Divide by the Denominator

Next, we need to divide the simplified expression by the denominator:

$$\frac{\square \cdot 3 \cdot x^{12} \cdot y^{3} \cdot z^{5}}{2 x^7 y^{\square} z^{\square}}$$

To divide by a variable with an exponent, we can subtract the exponent from the exponent of the numerator:

$$\frac{\square \cdot 3 \cdot x^{12-7} \cdot y^{3-\square} \cdot z^{5-\square}}{2}$$

Simplifying further, we get:

$$\frac{\square \cdot 3 \cdot x^5 \cdot y^{3-\square} \cdot z^{5-\square}}{2}$$

Step 3: Equate the Exponents

Now, we can equate the exponents of the variables on both sides of the equation:

$$x^5 \cdot y^{3-\square} \cdot z^{5-\square} = 12 x^{\square} y^9$$

Since the bases are the same, we can equate the exponents:

$$5 = \square$$

$$3-\square = 9$$

Solving for the second equation, we get:

$$\square = -6$$

Step 4: Solve for the Missing Exponents

Now that we have the values of the exponents, we can substitute them back into the original equation:

$$\frac{\left(-6 x^8 y^2 z^{-3}\right)\left(3 x^4 y^1 z^8\right)}{2 x^7 y^{-6} z^{-6}}=12 x^{5} y^9$$

Simplifying the left-hand side, we get:

$$\frac{-18 x^{12} y^3 z^5}{2 x^7 y^{-6} z^{-6}}=12 x^{5} y^9$$

Using the laws of exponents, we can simplify further:

$$\frac{-18 x^{12-7} y^{3+6} z^{5+6}}{2}=12 x^{5} y^9$$

Simplifying, we get:

$$\frac{-18 x^5 y^9 z^{11}}{2}=12 x^{5} y^9$$

Conclusion

In this article, we have solved an exponential equation by filling in the blanks to make the equation true. We started by simplifying the left-hand side of the equation, then divided by the denominator, and finally equated the exponents of the variables on both sides of the equation. We arrived at the solution by solving for the missing exponents and substituting them back into the original equation. This problem demonstrates the importance of understanding the laws of exponents and how to apply them to solve complex equations.

Final Answer

The final answer is:

$$\square = 5$$

$$\square = -6$$

\square = 11$<br/> **Frequently Asked Questions: Exponential Equations** =====================================================

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves variables with exponents. Exponents are used to represent repeated multiplication of a number by itself.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can start by combining the exponents using the laws of exponents. This involves adding or subtracting the exponents of the same base.

Q: What is the law of exponents?

A: The law of exponents states that when multiplying two numbers with the same base, you can add their exponents. When dividing two numbers with the same base, you can subtract their exponents.

Q: How do I divide an exponential expression by a variable with an exponent?

A: To divide an exponential expression by a variable with an exponent, you can subtract the exponent from the exponent of the numerator.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves variables with exponents, while a linear equation involves variables without exponents.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can start by simplifying the equation using the laws of exponents. Then, you can equate the exponents of the variables on both sides of the equation and solve for the missing exponents.

Q: What is the importance of understanding exponential equations?

A: Understanding exponential equations is important in many areas of mathematics and science, including algebra, calculus, and physics. Exponential equations are used to model real-world phenomena, such as population growth and chemical reactions.

Q: Can you provide an example of an exponential equation in real life?

A: Yes, an example of an exponential equation in real life is the growth of a population. If a population is growing at a rate of 2% per year, the number of people in the population can be modeled using an exponential equation.

Q: How do I determine the base of an exponential equation?

A: The base of an exponential equation is the number that is being raised to a power. To determine the base, you can look for the number that is being multiplied by itself repeatedly.

Q: Can you provide a step-by-step guide to solving an exponential equation?

A: Yes, here is a step-by-step guide to solving an exponential equation:

1. Simplify the equation using the laws of exponents.
2. Equate the exponents of the variables on both sides of the equation.
3. Solve for the missing exponents.
4. Substitute the values of the exponents back into the original equation.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not simplifying the equation using the laws of exponents.
• Not equating the exponents of the variables on both sides of the equation.
• Not solving for the missing exponents.
• Not substituting the values of the exponents back into the original equation.

Conclusion

In this article, we have answered some frequently asked questions about exponential equations. We have covered topics such as simplifying exponential equations, dividing exponential expressions, and solving exponential equations. We have also provided a step-by-step guide to solving exponential equations and discussed some common mistakes to avoid.