Simplify $\left(8 V^{\frac{1}{4}}\right)^{\frac{2}{3}}$.Provide Your Answer Below:

Introduction

In this article, we will simplify the given expression $\left(8 v^{\frac{1}{4}}\right)^{\frac{2}{3}}$. This involves applying the rules of exponents and simplifying the resulting expression.

Understanding the Expression

The given expression is $\left(8 v^{\frac{1}{4}}\right)^{\frac{2}{3}}$. This can be broken down into two parts: the base and the exponent.

• The base is $8 v^{\frac{1}{4}}$.
• The exponent is $\frac{2}{3}$.

Applying the Power Rule

The power rule states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.

Using this rule, we can simplify the given expression as follows:

$\left(8 v^{\frac{1}{4}}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}} \cdot \left(v^{\frac{1}{4}}\right)^{\frac{2}{3}}$

Simplifying the Base

The base $8$ can be simplified as follows:

$8^{\frac{2}{3}} = \left(2^3\right)^{\frac{2}{3}} = 2^2 = 4$

Simplifying the Exponent

The exponent $\left(v^{\frac{1}{4}}\right)^{\frac{2}{3}}$ can be simplified as follows:

$\left(v^{\frac{1}{4}}\right)^{\frac{2}{3}} = v^{\frac{1}{4} \cdot \frac{2}{3}} = v^{\frac{1}{6}}$

Combining the Simplified Base and Exponent

Now that we have simplified the base and exponent, we can combine them to get the final simplified expression:

$8^{\frac{2}{3}} \cdot \left(v^{\frac{1}{4}}\right)^{\frac{2}{3}} = 4 \cdot v^{\frac{1}{6}}$

Conclusion

In this article, we simplified the given expression $\left(8 v^{\frac{1}{4}}\right)^{\frac{2}{3}}$ using the power rule and simplifying the base and exponent. The final simplified expression is $4 \cdot v^{\frac{1}{6}}$.

Example Use Cases

This expression can be used in various mathematical contexts, such as:

• Simplifying complex expressions in algebra and calculus
• Solving equations and inequalities involving exponents
• Working with functions and their compositions

Tips and Tricks

When simplifying expressions involving exponents, remember to:

• Apply the power rule to simplify the expression
• Simplify the base and exponent separately
• Combine the simplified base and exponent to get the final simplified expression

Q: What is the power rule in mathematics?

A: The power rule is a fundamental rule in mathematics that states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions involving exponents.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, simply multiply the exponents of the base and the exponent. For example, if we have the expression $(2^3)^4$, we can simplify it by multiplying the exponents: $2^{3 \cdot 4} = 2^{12}$.

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

A: The base is the number or expression being raised to a power, while the exponent is the power to which the base is being raised. For example, in the expression $2^3$, the base is $2$ and the exponent is $3$.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by using the rule that $a^{-n} = \frac{1}{a^n}$. For example, if we have the expression $2^{-3}$, we can simplify it by using this rule: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

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

A: To simplify an expression with a fractional exponent, you can use the rule that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if we have the expression $2^{\frac{1}{2}}$, we can simplify it by using this rule: $2^{\frac{1}{2}} = \sqrt{2}$.

Q: Can I simplify an expression with a variable exponent?

A: Yes, you can simplify an expression with a variable exponent by using the power rule. For example, if we have the expression $(x^2)^3$, we can simplify it by multiplying the exponents: $x^{2 \cdot 3} = x^6$.

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

A: To simplify an expression with multiple bases and exponents, you can use the product rule, which states that $a^m \cdot b^n = a^m \cdot b^n$. For example, if we have the expression $2^3 \cdot 3^4$, we can simplify it by using this rule: $2^3 \cdot 3^4 = 8 \cdot 81 = 648$.

Q: Can I simplify an expression with a radical in the exponent?

A: Yes, you can simplify an expression with a radical in the exponent by using the rule that $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. For example, if we have the expression $\sqrt[3]{2^4}$, we can simplify it by using this rule: $\sqrt[3]{2^4} = 2^{\frac{4}{3}}$.

Conclusion

In this article, we have answered some frequently asked questions about simplifying expressions involving exponents. We have covered topics such as the power rule, simplifying expressions with negative and fractional exponents, and simplifying expressions with multiple bases and exponents. By following these tips and tricks, you can become more proficient in algebra and calculus.