What Is The Value Of $k$ In The Equation?$\[ \frac{20^2}{20^4} = 20^k \\]$k = \square$

Introduction

In mathematics, equations are used to represent relationships between variables. Solving for the value of a variable in an equation involves isolating that variable and finding its value. In this article, we will focus on solving for the value of k in the equation $\frac{20^2}{20^4} = 20^k$. We will use algebraic manipulation and properties of exponents to find the value of k.

Understanding the Equation

The given equation is $\frac{20^2}{20^4} = 20^k$. To solve for k, we need to simplify the left-hand side of the equation. We can do this by using the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$.

import math

# Define the variables
numerator = 20**2
denominator = 20**4

# Simplify the left-hand side of the equation
simplified_lhs = numerator / denominator

Using this property, we can simplify the left-hand side of the equation as follows:

$\frac{20^2}{20^4} = 20^{2-4} = 20^{-2}$

Solving for k

Now that we have simplified the left-hand side of the equation, we can equate it to the right-hand side and solve for k. We have:

$20^{-2} = 20^k$

To solve for k, we can use the property of exponents that states $a^m = a^n \implies m = n$. Applying this property to our equation, we get:

$-2 = k$

Conclusion

In this article, we solved for the value of k in the equation $\frac{20^2}{20^4} = 20^k$. We simplified the left-hand side of the equation using the property of exponents and then equated it to the right-hand side to solve for k. The value of k is $-2$.

Properties of Exponents

Exponents are a fundamental concept in mathematics, and they have several properties that can be used to simplify expressions and solve equations. Some of the key properties of exponents include:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
• Zero Exponent: $a^0 = 1$

These properties can be used to simplify expressions and solve equations involving exponents.

Examples of Solving Equations with Exponents

Here are a few examples of solving equations with exponents:

• Example 1: Solve for x in the equation $2^x = 16$.
• Solution: We can rewrite 16 as $2^4$, so the equation becomes $2^x = 2^4$. Using the property of exponents that states $a^m = a^n \implies m = n$, we get $x = 4$.
• Example 2: Solve for y in the equation $3^y = 81$.
• Solution: We can rewrite 81 as $3^4$, so the equation becomes $3^y = 3^4$. Using the property of exponents that states $a^m = a^n \implies m = n$, we get $y = 4$.

Real-World Applications of Exponents

Exponents have many real-world applications, including:

• Finance: Exponents are used to calculate compound interest and investment returns.
• Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems.
• Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: What is an exponent?

A: An exponent is a small number that is raised to a power, indicating how many times a base number should be multiplied by itself. For example, in the expression $2^3$, the 2 is the base and the 3 is the exponent.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you can use the following properties:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
• Zero Exponent: $a^0 = 1$

Q: How do I solve equations with exponents?

A: To solve equations with exponents, you can use the following steps:

1. Simplify the left-hand side of the equation using the properties of exponents.
2. Equate the simplified left-hand side to the right-hand side.
3. Use the property of exponents that states $a^m = a^n \implies m = n$ to solve for the variable.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. For example, in the expression $2^3$, the 2 is multiplied by itself 3 times.

A negative exponent indicates that the base number should be divided by itself a certain number of times. For example, in the expression $2^{-3}$, the 2 is divided by itself 3 times.

Q: How do I evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, you can use the following property:

$\frac{1}{a^m} = a^{-m}$

For example, to evaluate the expression $\frac{1}{2^3}$, you can rewrite it as $2^{-3}$ and then evaluate it as $\frac{1}{8}$.

Q: What is the value of $0^0$?

A: The value of $0^0$ is undefined. This is because any number raised to the power of 0 is equal to 1, but 0 raised to any power is equal to 0.

Q: How do I evaluate expressions with fractional exponents?

A: To evaluate expressions with fractional exponents, you can use the following property:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

For example, to evaluate the expression $2^{\frac{1}{2}}$, you can rewrite it as $\sqrt{2}$.

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is raised to a power, indicating how many times a base number should be multiplied by itself. A power is the result of raising a base number to an exponent.

For example, in the expression $2^3$, the 2 is the base and the 3 is the exponent. The result of raising 2 to the power of 3 is 8.

Q: How do I use exponents in real-world applications?

A: Exponents have many real-world applications, including:

• Finance: Exponents are used to calculate compound interest and investment returns.
• Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems.
• Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

In conclusion, exponents are a fundamental concept in mathematics, and they have many real-world applications. By understanding the properties of exponents and how to simplify and solve equations involving exponents, you can apply these concepts to a wide range of problems and situations.