What Is The Value Of The Expression Below?$81^{1 / 4}$A. 9 B. $\frac{81}{4}$ C. $ 9 4 \frac{9}{4} 4 9 [/tex] D. 3
Understanding the Problem
When dealing with exponents, it's essential to understand the rules and properties that govern them. In this case, we're given the expression $81^{1 / 4}$, and we need to determine its value. To do this, we'll need to apply the rules of exponents and simplify the expression.
The Rule of Fractional Exponents
A fractional exponent is a shorthand way of expressing a power that is a fraction. In general, if we have an expression of the form $a^{m/n}$, where $a$ is a number and $m$ and $n$ are integers, then we can rewrite it as $(am){1/n}$ or $\sqrt[n]{a^m}$.
Applying the Rule to the Given Expression
In our case, we have $81^{1 / 4}$. Using the rule of fractional exponents, we can rewrite this as $(811){1/4}$ or $\sqrt[4]{81^1}$.
Simplifying the Expression
Now that we've rewritten the expression, we can simplify it further. We know that $81 = 3^4$, so we can substitute this into our expression to get $(34){1/4}$ or $\sqrt[4]{(3^4)}$.
Evaluating the Expression
To evaluate this expression, we need to apply the rule of exponents that states $(am)n = a^{mn}$. In this case, we have $(34){1/4}$, which simplifies to $3^{4 \cdot (1/4)}$ or $3^1$.
Conclusion
Therefore, the value of the expression $81^{1 / 4}$ is $3^1$, which is equal to $3$.
Comparison with the Given Options
Let's compare our answer with the given options:
- A. 9: This is not correct, as $3^1$ is not equal to $9$.
- B. $\frac81}{4}${4}$.
- C. $\frac9}{4}${4}$.
- D. 3: This is the correct answer, as $3^1$ is equal to $3$.
Final Answer
The final answer is D. 3.
Q: What is the difference between a fractional exponent and a regular exponent?
A: A regular exponent is a whole number that is raised to a power, while a fractional exponent is a fraction that is raised to a power. For example, $2^3$ is a regular exponent, while $2^{1/2}$ is a fractional exponent.
Q: How do I simplify a fractional exponent?
A: To simplify a fractional exponent, you can use the rule of fractional exponents, which states that $a^{m/n} = (am){1/n}$ or $\sqrt[n]{a^m}$. For example, $2^{1/2}$ can be simplified to $\sqrt{2}$.
Q: What is the rule for multiplying exponents with the same base?
A: When multiplying exponents with the same base, you can add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.
Q: What is the rule for dividing exponents with the same base?
A: When dividing exponents with the same base, you can subtract the exponents. For example, $2^5 \div 2^3 = 2^{5-3} = 2^2$.
Q: How do I evaluate an expression with a fractional exponent?
A: To evaluate an expression with a fractional exponent, you can use the rule of fractional exponents and simplify the expression. For example, $81^{1/4}$ can be simplified to $3^1$, which is equal to $3$.
Q: What is the difference between a positive exponent and a negative exponent?
A: A positive exponent is a whole number or a fraction that is raised to a power, while a negative exponent is the reciprocal of a positive exponent. For example, $2^3$ is a positive exponent, while $2^{-3}$ is a negative exponent.
Q: How do I simplify a negative exponent?
A: To simplify a negative exponent, you can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3}$ can be simplified to $\frac{1}{2^3}$, which is equal to $\frac{1}{8}$.
Q: What is the rule for raising a power to a power?
A: When raising a power to a power, you can multiply the exponents. For example, $(23)4 = 2^{3 \cdot 4} = 2^{12}$.
Q: How do I evaluate an expression with multiple fractional exponents?
A: To evaluate an expression with multiple fractional exponents, you can use the rule of fractional exponents and simplify the expression. For example, $2^{1/2} \cdot 2^{1/3}$ can be simplified to $\sqrt{2} \cdot \sqrt[3]{2}$.
Q: What is the difference between a radical and a fractional exponent?
A: A radical is a symbol that represents a root, while a fractional exponent is a shorthand way of expressing a power that is a fraction. For example, $\sqrt{2}$ is a radical, while $2^{1/2}$ is a fractional exponent.
Q: How do I convert a radical to a fractional exponent?
A: To convert a radical to a fractional exponent, you can use the rule that $\sqrt[n]{a} = a^{1/n}$. For example, $\sqrt{2}$ can be converted to $2^{1/2}$.
Q: What is the rule for simplifying a product of radicals?
A: When simplifying a product of radicals, you can multiply the radicals. For example, $\sqrt{2} \cdot \sqrt{3}$ can be simplified to $\sqrt{2 \cdot 3}$, which is equal to $\sqrt{6}$.
Q: How do I simplify a quotient of radicals?
A: To simplify a quotient of radicals, you can divide the radicals. For example, $\sqrt{2} \div \sqrt{3}$ can be simplified to $\sqrt{2/3}$.
Q: What is the rule for simplifying a power of a radical?
A: When simplifying a power of a radical, you can multiply the exponents. For example, $(\sqrt{2})^4 = 2^{4/2} = 2^2$.
Q: How do I evaluate an expression with multiple radicals?
A: To evaluate an expression with multiple radicals, you can use the rules of radicals and simplify the expression. For example, $\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{4}$ can be simplified to $\sqrt{2 \cdot 3 \cdot 4}$, which is equal to $\sqrt{24}$.
Q: What is the difference between a rational exponent and a radical?
A: A rational exponent is a fraction that is raised to a power, while a radical is a symbol that represents a root. For example, $2^{1/2}$ is a rational exponent, while $\sqrt{2}$ is a radical.
Q: How do I convert a rational exponent to a radical?
A: To convert a rational exponent to a radical, you can use the rule that $a^{m/n} = \sqrt[n]{a^m}$. For example, $2^{1/2}$ can be converted to $\sqrt{2}$.
Q: What is the rule for simplifying a product of rational exponents?
A: When simplifying a product of rational exponents, you can multiply the exponents. For example, $2^{1/2} \cdot 2^{1/3}$ can be simplified to $2^{1/2 + 1/3}$.
Q: How do I simplify a quotient of rational exponents?
A: To simplify a quotient of rational exponents, you can divide the exponents. For example, $2^{1/2} \div 2^{1/3}$ can be simplified to $2^{1/2 - 1/3}$.
Q: What is the rule for simplifying a power of a rational exponent?
A: When simplifying a power of a rational exponent, you can multiply the exponents. For example, $(2{1/2})4 = 2^{4/2} = 2^2$.
Q: How do I evaluate an expression with multiple rational exponents?
A: To evaluate an expression with multiple rational exponents, you can use the rules of rational exponents and simplify the expression. For example, $2^{1/2} \cdot 2^{1/3} \cdot 2^{1/4}$ can be simplified to $2^{1/2 + 1/3 + 1/4}$.
Q: What is the difference between a radical and a rational exponent?
A: A radical is a symbol that represents a root, while a rational exponent is a fraction that is raised to a power. For example, $\sqrt{2}$ is a radical, while $2^{1/2}$ is a rational exponent.
Q: How do I convert a radical to a rational exponent?
A: To convert a radical to a rational exponent, you can use the rule that $\sqrt[n]{a} = a^{1/n}$. For example, $\sqrt{2}$ can be converted to $2^{1/2}$.
Q: What is the rule for simplifying a product of radicals and rational exponents?
A: When simplifying a product of radicals and rational exponents, you can multiply the radicals and rational exponents. For example, $\sqrt{2} \cdot 2^{1/3}$ can be simplified to $\sqrt{2} \cdot \sqrt[3]{2}$.
Q: How do I simplify a quotient of radicals and rational exponents?
A: To simplify a quotient of radicals and rational exponents, you can divide the radicals and rational exponents. For example, $\sqrt{2} \div 2^{1/3}$ can be simplified to $\sqrt{2} \div \sqrt[3]{2}$.
Q: What is the rule for simplifying a power of a radical and a rational exponent?
A: When simplifying a power of a radical and a rational exponent, you can multiply the exponents. For example, $(\sqrt{2})^4 = 2^{4/2} = 2^2$.
Q: How do I evaluate an expression with multiple radicals and rational exponents?
A: To evaluate an expression with multiple radicals and rational exponents, you can use the rules of radicals and rational exponents and simplify the expression. For example, $\sqrt{2} \cdot 2^{1/3} \cdot \sqrt{3}$ can be simplified to $\sqrt{2 \cdot 3} \cdot \sqrt[3]{2}$.
Q: What is the difference between a radical and a rational exponent in the context of real numbers?
A: In the context of real numbers, a radical is a symbol that represents a root, while a rational exponent is a fraction that is raised to a power. For example, $