Evaluate The Expression $2^{\frac{5}{2}}$.A. Unsolvable B. $2 \sqrt{2}$ C. $ 4 2 4 \sqrt{2} 4 2 [/tex] D. 8
Introduction
Exponential expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will focus on evaluating the expression $2^{\frac{5}{2}}$. We will break down the steps involved in evaluating this expression and provide a clear explanation of the process.
Understanding Exponential Expressions
Exponential expressions are written in the form $a^b$, where $a$ is the base and $b$ is the exponent. The exponent tells us how many times to multiply the base by itself. For example, $2^3$ means $2 \times 2 \times 2$, which equals $8$.
Evaluating the Expression $2^{\frac{5}{2}}$
To evaluate the expression $2^{\frac{5}{2}}$, we need to understand that the exponent $\frac{5}{2}$ means that we need to multiply the base $2$ by itself $\frac{5}{2}$ times.
Step 1: Understanding the Fractional Exponent
The exponent $\frac{5}{2}$ can be thought of as $2.5$. This means that we need to multiply the base $2$ by itself $2.5$ times.
Step 2: Multiplying the Base by Itself
To multiply the base $2$ by itself $2.5$ times, we can break it down into two parts: $2 \times 2$ and $2 \times 0.5$.
Step 3: Evaluating the First Part
The first part, $2 \times 2$, equals $4$.
Step 4: Evaluating the Second Part
The second part, $2 \times 0.5$, equals $1$.
Step 5: Combining the Results
Now, we need to combine the results of the first and second parts. We can do this by multiplying them together: $4 \times 1$, which equals $4$.
Step 6: Simplifying the Result
However, we are not done yet. We need to simplify the result by taking the square root of the base $2$ and multiplying it by the result we obtained in the previous step.
Step 7: Taking the Square Root
The square root of $2$ is $\sqrt{2}$.
Step 8: Multiplying the Result
Now, we need to multiply the result we obtained in the previous step by the square root of $2$: $4 \times \sqrt{2}$.
Step 9: Evaluating the Final Result
The final result is $4 \sqrt{2}$.
Conclusion
In conclusion, evaluating the expression $2^{\frac{5}{2}}$ requires a step-by-step approach. We need to understand the fractional exponent, multiply the base by itself, evaluate the first and second parts, combine the results, simplify the result, take the square root, and multiply the result. The final answer is $4 \sqrt{2}$.
Discussion
The correct answer is $4 \sqrt{2}$. This is because the expression $2^{\frac{5}{2}}$ can be evaluated by multiplying the base $2$ by itself $\frac{5}{2}$ times, which equals $4 \sqrt{2}$.
Comparison of Options
Let's compare the options:
- A. unsolvable: This is incorrect because the expression $2^{\frac{5}{2}}$ can be evaluated.
- B. $2 \sqrt2}${2}}$ equals $4 \sqrt{2}$, not $2 \sqrt{2}$.
- C. $4 \sqrt2}${2}}$ equals $4 \sqrt{2}$.
- D. 8: This is incorrect because the expression $2^{\frac{5}{2}}$ equals $4 \sqrt{2}$, not 8.
Final Answer
Introduction
In our previous article, we discussed how to evaluate the expression $2^{\frac{5}{2}}$. We broke down the steps involved in evaluating this expression and provided a clear explanation of the process. In this article, we will continue to explore exponential expressions and provide answers to some common questions.
Q&A
Q: What is an exponential expression?
A: An exponential expression is a mathematical expression that involves a base and an exponent. The base is the number that is being multiplied by itself, and the exponent is the number of times the base is multiplied by itself.
Q: How do I evaluate an exponential expression?
A: To evaluate an exponential expression, you need to follow these steps:
- Understand the base and the exponent.
- Multiply the base by itself as many times as indicated by the exponent.
- Simplify the result by combining like terms.
Q: What is a fractional exponent?
A: A fractional exponent is an exponent that is a fraction, such as $\frac{5}{2}$. When a fraction is used as an exponent, it means that the base is being multiplied by itself a fraction of the time.
Q: How do I evaluate an expression with a fractional exponent?
A: To evaluate an expression with a fractional exponent, you need to follow these steps:
- Understand the base and the fractional exponent.
- Multiply the base by itself as many times as indicated by the fractional exponent.
- Simplify the result by combining like terms.
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 and an exponent, while a polynomial expression is a mathematical expression that involves variables and coefficients. Exponential expressions involve repeated multiplication, while polynomial expressions involve addition and subtraction.
Q: Can I simplify an exponential expression?
A: Yes, you can simplify an exponential expression by combining like terms. For example, $2^3 \times 2^2$ can be simplified to $2^5$.
Q: How do I evaluate an expression with a negative exponent?
A: To evaluate an expression with a negative exponent, you need to follow these steps:
- Understand the base and the negative exponent.
- Take the reciprocal of the base and change the sign of the exponent.
- Multiply the base by itself as many times as indicated by the exponent.
Q: What is the order of operations for exponential expressions?
A: The order of operations for exponential expressions is:
- Evaluate any expressions inside parentheses.
- Evaluate any exponential expressions.
- Multiply and divide from left to right.
- Add and subtract from left to right.
Common Mistakes
Mistake 1: Not understanding the base and the exponent
A: Make sure you understand the base and the exponent before evaluating the expression.
Mistake 2: Not following the order of operations
A: Make sure you follow the order of operations when evaluating an exponential expression.
Mistake 3: Not simplifying the result
A: Make sure you simplify the result by combining like terms.
Conclusion
In conclusion, evaluating exponential expressions requires a clear understanding of the base and the exponent. By following the steps outlined in this article, you can evaluate exponential expressions with ease. Remember to simplify the result by combining like terms and follow the order of operations.
Final Tips
- Make sure you understand the base and the exponent before evaluating the expression.
- Follow the order of operations when evaluating an exponential expression.
- Simplify the result by combining like terms.
Common Exponential Expressions
Practice Problems
- Evaluate the expression $2^{\frac{3}{2}}$.
- Evaluate the expression $3^{\frac{5}{3}}$.
- Evaluate the expression $4^{-2}$.