What Is The Value Of The Expression Below? 81 5 / 4 81^{5 / 4} 8 1 5/4 A. 15 B. 3 C. 105 D. 243

Mar 10, 2025 by ADMIN 99 views

What is the Value of the Expression Below?

The expression $81^{5 / 4}$ may seem complex at first glance, but it can be simplified using the properties of exponents. In this article, we will break down the expression and calculate its value.

Understanding Exponents

Before we dive into the expression, let's quickly review what exponents are. An exponent is a small number that is raised to the power of a larger number. For example, in the expression $2^3$ , the 3 is the exponent and the 2 is the base. The value of the expression is calculated by multiplying the base by itself as many times as the exponent indicates. In this case, $2^3 = 2 \times 2 \times 2 = 8$ .

Simplifying the Expression

Now that we understand exponents, let's simplify the expression $81^{5 / 4}$ . The first step is to recognize that 81 is a perfect square, which means it can be expressed as a power of a smaller number. Specifically, $81 = 9^2$ . We can rewrite the expression as $(9^2)^{5 / 4}$ .

Using the Power of a Power Rule

The next step is to apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$ . In this case, we have $(9^2)^{5 / 4} = 9^{2 \times 5 / 4}$ . Simplifying the exponent, we get $9^{10 / 4}$ .

Simplifying the Exponent

Now that we have $9^{10 / 4}$ , we can simplify the exponent further. The fraction $10 / 4$ can be reduced to $5 / 2$ . Therefore, the expression becomes $9^{5 / 2}$ .

Using the Power of a Fraction Rule

The power of a fraction rule states that $a^{m / n} = (a^m)^{1 / n}$ . In this case, we have $9^{5 / 2} = (9^5)^{1 / 2}$ . Simplifying the expression, we get $(9^5)^{1 / 2}$ .

Calculating the Value

Now that we have $(9^5)^{1 / 2}$ , we can calculate the value of the expression. The first step is to calculate $9^5$ . This is equal to $9 \times 9 \times 9 \times 9 \times 9 = 59049$ . Therefore, $(9^5)^{1 / 2} = 59049^{1 / 2}$ .

Simplifying the Square Root

The final step is to simplify the square root of 59049. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of 59049 is 243, because $243 \times 243 = 59049$ .

Conclusion

Answer

The correct answer is D. 243.

Additional Tips and Tricks

When simplifying expressions with exponents, it's essential to recognize perfect squares and cubes, as these can be expressed as powers of smaller numbers.
The power of a power rule and the power of a fraction rule are essential tools for simplifying expressions with exponents.
When calculating the value of an expression, it's crucial to follow the order of operations (PEMDAS), which includes parentheses, exponents, multiplication and division, and addition and subtraction.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's easy to get confused and make mistakes. To avoid this, make sure to follow the order of operations and use the correct rules for simplifying expressions.
Don't forget to simplify the exponent before calculating the value of the expression.
When calculating the value of an expression, make sure to follow the order of operations and use the correct rules for simplifying expressions.

Real-World Applications

Exponents are used extensively in mathematics, science, and engineering to describe complex relationships and patterns.
Understanding exponents and how to simplify expressions with them is essential for solving problems in fields such as physics, chemistry, and computer science.
Exponents are also used in finance and economics to describe interest rates and investment returns.

Conclusion

In conclusion, the value of the expression $81^{5 / 4}$ is 243. This was achieved by simplifying the expression using the properties of exponents, including the power of a power rule, the power of a fraction rule, and the square root of a number. Understanding exponents and how to simplify expressions with them is essential for solving problems in mathematics, science, and engineering.
Q&A: Understanding Exponents and Simplifying Expressions

In the previous article, we explored the value of the expression $81^{5 / 4}$ and simplified it using the properties of exponents. In this article, we will answer some common questions related to exponents and simplifying expressions.

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

A: The base is the number that is being raised to a power, while the exponent is the small number that indicates how many times the base is multiplied by itself.

Q: How do I simplify an expression with a fraction as an exponent?

A: To simplify an expression with a fraction as an exponent, you can use the power of a fraction rule, which states that $a^{m / n} = (a^m)^{1 / n}$ . This rule allows you to rewrite the fraction as a power of a smaller number.

Q: What is the power of a power rule?

A: The power of a power rule states that $(a^m)^n = a^{m \times n}$ . This rule allows you to simplify expressions with multiple exponents by multiplying the exponents together.

Q: How do I calculate the value of an expression with a square root?

A: To calculate the value of an expression with a square root, you can use the fact that the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$ .

Q: What is the difference between a perfect square and a perfect cube?

A: A perfect square is a number that can be expressed as a power of 2, while a perfect cube is a number that can be expressed as a power of 3. For example, 16 is a perfect square because it can be expressed as $4^2$ , while 27 is a perfect cube because it can be expressed as $3^3$ .

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the power of a power rule and the power of a fraction rule. For example, if you have the expression $(2^3)^4$ , you can simplify it by using the power of a power rule to get $2^{3 \times 4} = 2^{12}$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is:

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

Q: Why is it important to follow the order of operations?

A: Following the order of operations is essential to ensure that mathematical expressions are evaluated correctly. If you don't follow the order of operations, you may get incorrect results.

Q: Can you provide some examples of real-world applications of exponents?

A: Yes, exponents are used extensively in mathematics, science, and engineering to describe complex relationships and patterns. Some examples of real-world applications of exponents include:

Calculating interest rates and investment returns in finance and economics.
Describing the growth and decay of populations in biology and ecology.
Modeling the behavior of complex systems in physics and engineering.
Analyzing data and making predictions in statistics and data science.

Conclusion

In conclusion, understanding exponents and simplifying expressions is essential for solving problems in mathematics, science, and engineering. By following the power of a power rule, the power of a fraction rule, and the order of operations, you can simplify complex expressions and calculate their values with ease.