Evaluating the First Part of the Expression

Oct 13, 2025 by windailys.com 122 views

Introduction

In mathematics, evaluating expressions is a crucial skill that involves simplifying mathematical expressions by applying the rules of arithmetic operations. In this article, we will focus on evaluating the expression $4 \times 4^{\frac{5}{2}} + 8^{\frac{1}{3}}$ . This expression involves exponents, multiplication, and addition, making it a challenging problem to solve. We will break down the expression into smaller parts, apply the rules of exponents, and simplify the expression step by step.

Understanding Exponents

Before we start evaluating the expression, let's review the rules of exponents. An exponent is a small number that is written to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, $4^3$ means $4$ multiplied by itself $3$ times, which is equal to $64$ . In general, $a^b$ means $a$ multiplied by itself $b$ times.

Evaluating the First Part of the Expression

The first part of the expression is $4 \times 4^{\frac{5}{2}}$ . To evaluate this part, we need to apply the rule of exponents that states $a^m \times a^n = a^{m+n}$ . In this case, we have $4^{\frac{5}{2}}$ , which can be rewritten as $(4^{\frac{1}{2}})^5$ . Since $4^{\frac{1}{2}}$ is equal to $2$ , we can simplify the expression as follows:

4 \times 4^{\frac{5}{2}} = 4 \times (2)^5 = 4 \times 32 = 128

Evaluating the Second Part of the Expression

The second part of the expression is $8^{\frac{1}{3}}$ . To evaluate this part, we need to apply the rule of exponents that states $(a^m)^n = a^{m \times n}$ . In this case, we have $8^{\frac{1}{3}}$ , which can be rewritten as $(2^3)^{\frac{1}{3}}$ . Since $(2^3)^{\frac{1}{3}}$ is equal to $2$ , we can simplify the expression as follows:

8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2

Combining the Two Parts of the Expression

Now that we have evaluated the two parts of the expression, we can combine them to get the final result. The expression is $4 \times 4^{\frac{5}{2}} + 8^{\frac{1}{3}}$ , which we have simplified to $128 + 2$ . Therefore, the final result is:

4 \times 4^{\frac{5}{2}} + 8^{\frac{1}{3}} = 128 + 2 = 130

Conclusion

In this article, we evaluated the expression $4 \times 4^{\frac{5}{2}} + 8^{\frac{1}{3}}$ by breaking it down into smaller parts, applying the rules of exponents, and simplifying the expression step by step. We found that the final result is $130$ . This problem required a good understanding of exponents and the rules of arithmetic operations, making it a challenging problem to solve. However, by following the steps outlined in this article, we were able to simplify the expression and arrive at the final result.

Frequently Asked Questions

What is the rule of exponents that states $a^m \times a^n = a^{m+n}$ ? The rule of exponents that states $a^m \times a^n = a^{m+n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents.
How do we evaluate the expression $4 \times 4^{\frac{5}{2}}$ ? To evaluate the expression $4 \times 4^{\frac{5}{2}}$ , we need to apply the rule of exponents that states $a^m \times a^n = a^{m+n}$ . In this case, we have $4^{\frac{5}{2}}$ , which can be rewritten as $(4^{\frac{1}{2}})^5$ . Since $4^{\frac{1}{2}}$ is equal to $2$ , we can simplify the expression as follows: $4 \times 4^{\frac{5}{2}} = 4 \times (2)^5 = 4 \times 32 = 128$ .
How do we evaluate the expression $8^{\frac{1}{3}}$ ? To evaluate the expression $8^{\frac{1}{3}}$ , we need to apply the rule of exponents that states $(a^m)^n = a^{m \times n}$ . In this case, we have $8^{\frac{1}{3}}$ , which can be rewritten as $(2^3)^{\frac{1}{3}}$ . Since $(2^3)^{\frac{1}{3}}$ is equal to $2$ , we can simplify the expression as follows: $8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2$ .

Final Answer

The final answer is: $\boxed{130}$

Introduction

In our previous article, we evaluated the expression $4 \times 4^{\frac{5}{2}} + 8^{\frac{1}{3}}$ by breaking it down into smaller parts, applying the rules of exponents, and simplifying the expression step by step. In this article, we will answer some frequently asked questions related to evaluating expressions with exponents.

Q&A

Q: What is the rule of exponents that states $a^m \times a^n = a^{m+n}$ ?

A: The rule of exponents that states $a^m \times a^n = a^{m+n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents. This rule states that when we multiply two numbers with the same base, we can add their exponents.

Q: How do we evaluate the expression $4 \times 4^{\frac{5}{2}}$ ?

A: To evaluate the expression $4 \times 4^{\frac{5}{2}}$ , we need to apply the rule of exponents that states $a^m \times a^n = a^{m+n}$ . In this case, we have $4^{\frac{5}{2}}$ , which can be rewritten as $(4^{\frac{1}{2}})^5$ . Since $4^{\frac{1}{2}}$ is equal to $2$ , we can simplify the expression as follows: $4 \times 4^{\frac{5}{2}} = 4 \times (2)^5 = 4 \times 32 = 128$ .

Q: How do we evaluate the expression $8^{\frac{1}{3}}$ ?

A: To evaluate the expression $8^{\frac{1}{3}}$ , we need to apply the rule of exponents that states $(a^m)^n = a^{m \times n}$ . In this case, we have $8^{\frac{1}{3}}$ , which can be rewritten as $(2^3)^{\frac{1}{3}}$ . Since $(2^3)^{\frac{1}{3}}$ is equal to $2$ , we can simplify the expression as follows: $8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2$ .

Q: What is the difference between $a^m$ and $a^n$ ?

A: $a^m$ and $a^n$ are two different expressions with the same base $a$ . The exponent $m$ is the power to which the base $a$ is raised, while the exponent $n$ is the power to which the base $a$ is raised. For example, $2^3$ and $2^4$ are two different expressions with the same base $2$ .

Q: How do we simplify the expression $4^{\frac{1}{2}}$ ?

A: To simplify the expression $4^{\frac{1}{2}}$ , we need to find the square root of $4$ . Since the square root of $4$ is $2$ , we can simplify the expression as follows: $4^{\frac{1}{2}} = 2$ .

Q: What is the rule of exponents that states $(a^m)^n = a^{m \times n}$ ?

A: The rule of exponents that states $(a^m)^n = a^{m \times n}$ is a fundamental rule in mathematics that allows us to simplify expressions involving exponents. This rule states that when we raise a power to a power, we can multiply the exponents.

Q: How do we evaluate the expression $2^3 \times 2^4$ ?

A: To evaluate the expression $2^3 \times 2^4$ , we need to apply the rule of exponents that states $a^m \times a^n = a^{m+n}$ . In this case, we have $2^3$ and $2^4$ , which can be rewritten as $2^{3+4}$ . Since $2^{3+4}$ is equal to $2^7$ , we can simplify the expression as follows: $2^3 \times 2^4 = 2^7$ .

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions with exponents. We reviewed the rules of exponents, including the rule that states $a^m \times a^n = a^{m+n}$ and the rule that states $(a^m)^n = a^{m \times n}$ . We also provided examples of how to simplify expressions involving exponents. By following the steps outlined in this article, you should be able to evaluate expressions with exponents with confidence.

Final Answer

The final answer is: $\boxed{130}$