Understanding Exponents and Variables

Oct 13, 2025 by windailys.com 15 views

Solving for the Value of k in an Algebraic Equation

In algebra, solving for the value of a variable is a crucial step in understanding and working with equations. In this article, we will explore how to solve for the value of k in the given equation: $\left(5 a^2 b^3\right)\left(6 a^k b\right) = 30 a^6 b^4$ . This equation involves the use of exponents and variables, making it a great example of how to apply algebraic principles to solve for an unknown value.

Understanding Exponents and Variables

Before we dive into solving for k, it's essential to understand the basics of exponents and variables. Exponents represent the power to which a number is raised, while variables are letters or symbols that represent unknown values. In the given equation, we have exponents on both sides of the equation, which we will need to use to solve for k.

Applying the Laws of Exponents

To solve for k, we need to apply the laws of exponents. The law of exponents states that when multiplying two numbers with the same base, we add their exponents. In this case, we have:

$\left(5 a^2 b^3\right)\left(6 a^k b\right) = 30 a^6 b^4$

Using the law of exponents, we can rewrite the left-hand side of the equation as:

$5 \cdot 6 \cdot a^{2+k} \cdot b^{3+1} = 30 a^6 b^4$

Simplifying the Equation

Now that we have applied the law of exponents, we can simplify the equation by combining like terms. We can rewrite the equation as:

$30 a^{2+k} b^4 = 30 a^6 b^4$

Equating Exponents

Since the bases on both sides of the equation are the same (a and b), we can equate the exponents. This gives us:

$2+k = 6$

Solving for k

Now that we have equated the exponents, we can solve for k by subtracting 2 from both sides of the equation:

$k = 6 - 2$

$k = 4$

In conclusion, we have solved for the value of k in the given equation: $\left(5 a^2 b^3\right)\left(6 a^k b\right) = 30 a^6 b^4$ . By applying the laws of exponents and simplifying the equation, we found that the value of k is 4. This is a great example of how to use algebraic principles to solve for an unknown value.

The correct answer is C. 4.

Here are a few additional examples of how to solve for the value of k in similar equations:

$\left(2 a^3 b^2\right)\left(4 a^k b\right) = 8 a^6 b^3$
$\left(3 a^2 b^4\right)\left(6 a^k b\right) = 18 a^6 b^5$
$\left(4 a^3 b^2\right)\left(8 a^k b\right) = 32 a^6 b^4$

In each of these examples, we can apply the laws of exponents and simplify the equation to solve for the value of k.
Frequently Asked Questions (FAQs) about Solving for k in Algebraic Equations

In our previous article, we explored how to solve for the value of k in the equation: $\left(5 a^2 b^3\right)\left(6 a^k b\right) = 30 a^6 b^4$ . We applied the laws of exponents and simplified the equation to find that the value of k is 4. In this article, we will answer some frequently asked questions (FAQs) about solving for k in algebraic equations.

Q: What is the law of exponents?

A: The law of exponents states that when multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$ .

Q: How do I apply the law of exponents to solve for k?

A: To apply the law of exponents, you need to multiply the coefficients (numbers in front of the variables) and add the exponents of the variables. For example, in the equation $\left(5 a^2 b^3\right)\left(6 a^k b\right) = 30 a^6 b^4$ , we multiply the coefficients (5 and 6) and add the exponents of the variables (2 and k).

Q: What if the bases are different?

A: If the bases are different, you cannot apply the law of exponents directly. However, you can still simplify the equation by multiplying the coefficients and adding the exponents of the variables. For example, in the equation $\left(5 a^2 b^3\right)\left(6 c^k d\right) = 30 a^6 b^4 c^k d$ , we multiply the coefficients (5 and 6) and add the exponents of the variables (2 and k).

Q: How do I simplify the equation after applying the law of exponents?

A: After applying the law of exponents, you can simplify the equation by combining like terms. For example, in the equation $5 \cdot 6 \cdot a^{2+k} \cdot b^{3+1} = 30 a^6 b^4$ , we can simplify the equation by combining like terms: $30 a^{2+k} b^4 = 30 a^6 b^4$ .

Q: What if the equation has multiple variables?

A: If the equation has multiple variables, you can still apply the law of exponents and simplify the equation. For example, in the equation $\left(5 a^2 b^3 c^4\right)\left(6 a^k b^2 c^3\right) = 30 a^6 b^4 c^7$ , we can apply the law of exponents and simplify the equation: $30 a^{2+k} b^{3+2} c^{4+3} = 30 a^6 b^4 c^7$ .

Q: Can I use the law of exponents to solve for k in a quadratic equation?

A: Yes, you can use the law of exponents to solve for k in a quadratic equation. However, you need to be careful when applying the law of exponents to quadratic equations. For example, in the equation $x^2 + 5x + 6 = 0$ , we can apply the law of exponents to solve for k: $x^2 + 5x + 6 = (x+2)(x+3) = 0$ .

In conclusion, solving for k in algebraic equations involves applying the law of exponents and simplifying the equation. By following these steps, you can solve for k in a variety of equations, including those with multiple variables and quadratic equations. Remember to be careful when applying the law of exponents, and don't hesitate to ask for help if you need it.

If you need additional help or resources, here are some suggestions:

Khan Academy: Algebra
Mathway: Algebra Solver
Wolfram Alpha: Algebra Calculator

The correct answers to the FAQs are:

Q: What is the law of exponents? A: The law of exponents states that when multiplying two numbers with the same base, we add their exponents.
Q: How do I apply the law of exponents to solve for k? A: To apply the law of exponents, you need to multiply the coefficients and add the exponents of the variables.
Q: What if the bases are different? A: If the bases are different, you cannot apply the law of exponents directly.
Q: How do I simplify the equation after applying the law of exponents? A: After applying the law of exponents, you can simplify the equation by combining like terms.
Q: What if the equation has multiple variables? A: If the equation has multiple variables, you can still apply the law of exponents and simplify the equation.
Q: Can I use the law of exponents to solve for k in a quadratic equation? A: Yes, you can use the law of exponents to solve for k in a quadratic equation.