Find The Value Of (-3)^4 (5/3) ^4

Feb 28, 2025 by ADMIN 35 views

Solving Exponential Expressions: A Step-by-Step Guide to Finding the Value of (-3)^4 (5/3) ^4

Exponential expressions are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various mathematical disciplines. In this article, we will delve into the world of exponential expressions and provide a step-by-step guide on how to find the value of (-3)^4 (5/3) ^4. We will explore the properties of exponents, learn how to simplify complex expressions, and ultimately arrive at the solution.

Before we dive into the solution, it's essential to understand the basics of exponents. An exponent is a small number that is raised to a power, indicating how many times the base number should be multiplied by itself. For example, in the expression 2^3, the base number is 2, and the exponent is 3. This means that 2 should be multiplied by itself 3 times, resulting in 2 × 2 × 2 = 8.

There are several properties of exponents that we need to be aware of when solving exponential expressions. These properties include:

Product of Powers: When multiplying two or more numbers with the same base, we can add the exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
Power of a Power: When raising a power to another power, we can multiply the exponents. For example, (2³⁾4 = 2^(3×4) = 2^12.
Zero Exponent: Any number raised to the power of 0 is equal to 1. For example, 2^0 = 1.

Now that we have a solid understanding of exponents and their properties, let's simplify the expression (-3)^4 (5/3) ^4.

Step 1: Simplify the First Part of the Expression

The first part of the expression is (-3)^4. Using the property of exponents, we can simplify this as follows:

(-3)^4 = (-3) × (-3) × (-3) × (-3) = 81

Step 2: Simplify the Second Part of the Expression

The second part of the expression is (5/3) ^4. Using the property of exponents, we can simplify this as follows:

(5/3) ^4 = (5/3) × (5/3) × (5/3) × (5/3) = 625/81

Step 3: Multiply the Two Simplified Expressions

Now that we have simplified both parts of the expression, we can multiply them together:

81 × (625/81) = 625

In this article, we have learned how to simplify complex exponential expressions using the properties of exponents. We have applied these properties to the expression (-3)^4 (5/3) ^4 and arrived at the solution of 625. By understanding the basics of exponents and their properties, we can tackle even the most complex mathematical problems with confidence.

When working with exponential expressions, it's essential to avoid common mistakes that can lead to incorrect solutions. Some of these mistakes include:

Forgetting to apply the properties of exponents: Make sure to apply the properties of exponents, such as the product of powers and power of a power, to simplify complex expressions.
Not simplifying the expression correctly: Take the time to simplify the expression correctly, using the properties of exponents and the order of operations.
Not checking the solution: Always check the solution to ensure that it is correct and makes sense in the context of the problem.

Exponential expressions have numerous real-world applications in fields such as science, engineering, and economics. Some examples include:

Population growth: Exponential expressions can be used to model population growth, where the population increases at a rate proportional to the current population.
Compound interest: Exponential expressions can be used to calculate compound interest, where the interest earned is added to the principal amount and the resulting amount is then multiplied by the interest rate.
Radioactive decay: Exponential expressions can be used to model radioactive decay, where the amount of radioactive material decreases at a rate proportional to the current amount.

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that involves a base number raised to a power. For example, 2^3 is an exponential expression where 2 is the base and 3 is the exponent.

Q: What are the properties of exponents?

A: The properties of exponents include:

Product of Powers: When multiplying two or more numbers with the same base, we can add the exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
Power of a Power: When raising a power to another power, we can multiply the exponents. For example, (2³⁾4 = 2^(3×4) = 2^12.
Zero Exponent: Any number raised to the power of 0 is equal to 1. For example, 2^0 = 1.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can use the properties of exponents. For example, to simplify 2^3 × 2^4, you can add the exponents: 2^(3+4) = 2^7.

Q: What is the order of operations for exponential expressions?

A: The order of operations for exponential expressions is the same as for regular mathematical expressions:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an exponential expression with a negative base?

A: To evaluate an exponential expression with a negative base, you can use the property of exponents that states (-a)^n = a^n if n is even, and (-a)^n = -a^n if n is odd. For example, (-2)^3 = -8.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is a mathematical expression that involves a base number raised to a power, while a polynomial expression is a mathematical expression that involves a sum of terms, each of which is a product of a coefficient and a variable raised to a power. For example, 2^3 is an exponential expression, while 2x^2 + 3x + 1 is a polynomial expression.

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

A: Exponential expressions have numerous real-world applications in fields such as science, engineering, and economics. Some examples include:

Population growth: Exponential expressions can be used to model population growth, where the population increases at a rate proportional to the current population.
Compound interest: Exponential expressions can be used to calculate compound interest, where the interest earned is added to the principal amount and the resulting amount is then multiplied by the interest rate.
Radioactive decay: Exponential expressions can be used to model radioactive decay, where the amount of radioactive material decreases at a rate proportional to the current amount.

Q: What are some common mistakes to avoid when working with exponential expressions?

A: Some common mistakes to avoid when working with exponential expressions include:

Forgetting to apply the properties of exponents: Make sure to apply the properties of exponents, such as the product of powers and power of a power, to simplify complex expressions.
Not simplifying the expression correctly: Take the time to simplify the expression correctly, using the properties of exponents and the order of operations.
Not checking the solution: Always check the solution to ensure that it is correct and makes sense in the context of the problem.

Q: How can I practice working with exponential expressions?

A: There are many ways to practice working with exponential expressions, including:

Solving problems: Try solving problems that involve exponential expressions, such as those found in math textbooks or online resources.
Using online tools: Use online tools, such as calculators or math software, to practice working with exponential expressions.
Working with a tutor or teacher: Work with a tutor or teacher who can provide guidance and support as you practice working with exponential expressions.