Evaluate The Expression $16^{-\frac{3}{2}}$.
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Introduction
In mathematics, expressions involving exponents and fractions can be challenging to evaluate. The given expression $16^{-\frac{3}{2}}$ is a perfect example of such a problem. In this article, we will break down the expression and provide a step-by-step solution to evaluate it.
Understanding the Expression
The given expression is $16^{-\frac{3}{2}}$. To evaluate this expression, we need to understand the properties of exponents and fractions. The exponent $-\frac{3}{2}$ indicates that we need to take the reciprocal of the base (16) and raise it to the power of $-\frac{3}{2}$.
Breaking Down the Expression
To simplify the expression, we can break it down into smaller parts. We can rewrite the expression as $\frac{1}{16^{\frac{3}{2}}}$.
Evaluating the Exponent
Now, let's focus on evaluating the exponent $\frac{3}{2}$. We can rewrite 16 as $2^4$, since $2^4 = 16$. Therefore, we can rewrite the expression as $\frac{1}{(24){\frac{3}{2}}}$.
Applying the Power Rule
The power rule states that $(am)n = a^{mn}$. We can apply this rule to simplify the expression. Therefore, we can rewrite the expression as $\frac{1}{2^{4 \cdot \frac{3}{2}}}$.
Simplifying the Expression
Now, let's simplify the expression further. We can multiply the exponent $4 \cdot \frac{3}{2}$ to get $6$. Therefore, we can rewrite the expression as $\frac{1}{2^6}$.
Evaluating the Final Expression
The final expression is $\frac{1}{2^6}$. To evaluate this expression, we can rewrite it as $\frac{1}{64}$.
Conclusion
In conclusion, we have successfully evaluated the expression $16^{-\frac{3}{2}}$. We broke down the expression into smaller parts, applied the power rule, and simplified the expression to get the final answer $\frac{1}{64}$.
Frequently Asked Questions
Q: What is the value of $16^{-\frac{3}{2}}$?
A: The value of $16^{-\frac{3}{2}}$ is $\frac{1}{64}$.
Q: How do I evaluate an expression with a negative exponent?
A: To evaluate an expression with a negative exponent, you need to take the reciprocal of the base and raise it to the power of the negative exponent.
Q: What is the power rule in mathematics?
A: The power rule states that $(am)n = a^{mn}$.
Final Answer
The final answer is $\frac{1}{64}$.
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Introduction
In our previous article, we evaluated the expression $16^{-\frac{3}{2}}$. In this article, we will provide a Q&A section to help you better understand how to evaluate expressions with exponents and fractions.
Q&A
Q: What is the value of $2^{-3}$?
A: To evaluate this expression, we need to take the reciprocal of the base (2) and raise it to the power of -3. Therefore, we can rewrite the expression as $\frac{1}{2^3}$. Since $2^3 = 8$, the value of $2^{-3}$ is $\frac{1}{8}$.
Q: How do I evaluate an expression with a negative exponent and a fraction?
A: To evaluate an expression with a negative exponent and a fraction, you need to follow these steps:
- Take the reciprocal of the base.
- Raise the reciprocal to the power of the negative exponent.
- Simplify the expression.
For example, let's evaluate the expression $\frac{1}{2^{-3}}$. We can rewrite the expression as $2^3$, since the reciprocal of $2^{-3}$ is $2^3$. Therefore, the value of $\frac{1}{2^{-3}}$ is $2^3 = 8$.
Q: What is the power rule in mathematics?
A: The power rule states that $(am)n = a^{mn}$. This rule can be used to simplify expressions with exponents.
Q: How do I apply the power rule to an expression?
A: To apply the power rule to an expression, you need to multiply the exponents. For example, let's evaluate the expression $(23)4$. We can rewrite the expression as $2^{3 \cdot 4}$, since the power rule states that $(am)n = a^{mn}$. Therefore, the value of $(23)4$ is $2^{12}$.
Q: What is the difference between a positive exponent and a negative exponent?
A: A positive exponent indicates that you need to multiply the base by itself as many times as the exponent. A negative exponent indicates that you need to take the reciprocal of the base and raise it to the power of the absolute value of the exponent.
Q: How do I evaluate an expression with a fraction and a negative exponent?
A: To evaluate an expression with a fraction and a negative exponent, you need to follow these steps:
- Take the reciprocal of the base.
- Raise the reciprocal to the power of the negative exponent.
- Simplify the expression.
For example, let's evaluate the expression $\frac{1}{2^{-3}}$. We can rewrite the expression as $2^3$, since the reciprocal of $2^{-3}$ is $2^3$. Therefore, the value of $\frac{1}{2^{-3}}$ is $2^3 = 8$.
Common Mistakes
Mistake 1: Not following the order of operations
When evaluating expressions with exponents and fractions, it's essential to follow the order of operations (PEMDAS):
- Parentheses
- Exponents
- Multiplication and Division
- Addition and Subtraction
Mistake 2: Not simplifying the expression
When evaluating expressions with exponents and fractions, it's essential to simplify the expression as much as possible.
Mistake 3: Not using the power rule
When evaluating expressions with exponents, it's essential to use the power rule to simplify the expression.
Conclusion
In conclusion, evaluating expressions with exponents and fractions can be challenging, but with practice and patience, you can master these skills. Remember to follow the order of operations, simplify the expression, and use the power rule to evaluate expressions with exponents and fractions.
Final Answer
The final answer is $\frac{1}{64}$.