The first step is to identify the base and the exponent inside the parentheses

Oct 13, 2025 by thesweetdesigns.com 122 views

Understanding the Problem

When dealing with exponents, it's essential to understand the rules of exponentiation. The given expression involves a power raised to another power, which can be simplified using the power rule of exponents. The power rule states that for any variables a and b and any integers m and n, (a^m)n = a^(m*n). This rule will be applied to simplify the given expression.

Breaking Down the Expression

The given expression is $\left(100 v^{\frac{2}{5}}\right)^{\frac{3}{2}}$ . To simplify this, we need to apply the power rule of exponents. The first step is to identify the base and the exponent inside the parentheses. In this case, the base is $100 v^{\frac{2}{5}}$ and the exponent is $\frac{3}{2}$ .

Applying the Power Rule

Using the power rule, we can rewrite the expression as $100 v^{\frac{2}{5}}$ raised to the power of $\frac{3}{2}$ . This can be written as (100 v^{{\frac{2}{5}})}{\frac{3}{2}} = 100^{\frac{3}{2}} * (v^{{\frac{2}{5}})}{\frac{3}{2}}$.

Simplifying the Expression

Now, we can simplify the expression by applying the power rule to the variables. For the variable part, we have (v^{{\frac{2}{5}})}{\frac{3}{2}} = v^{\frac{2}{5} * \frac{3}{2}} = v^{\frac{3}{5}}$. This simplifies the expression to $100^{\frac{3}{2}} * v^{\frac{3}{5}}$ .

Evaluating the Coefficients

The next step is to evaluate the coefficients. The coefficient 100 can be rewritten as 10^2, which makes it easier to evaluate the exponent. Using the property of exponents that states (a^m)n = a^(m*n), we can rewrite $100^{\frac{3}{2}}$ as $(10^2)^{\frac{3}{2}} = 10^{2 * \frac{3}{2}} = 10^3$ .

Final Simplification

Now, we can simplify the expression by combining the coefficients and the variables. The final simplified expression is 10^3 * v^{\frac{3}{5}} = 1000 v^{\frac{3}{5}}$.

Conclusion

In this article, we simplified the expression $\left(100 v^{\frac{2}{5}}\right)^{\frac{3}{2}}$ using the power rule of exponents. The final simplified expression is 1000 v^{\frac{3}{5}}$, which demonstrates the application of the power rule in simplifying expressions involving exponents.

Frequently Asked Questions

Q: What is the power rule of exponents? A: The power rule of exponents states that for any variables a and b and any integers m and n, (a^m)n = a^(m*n).
Q: How do you apply the power rule to simplify an expression? A: To apply the power rule, identify the base and the exponent inside the parentheses, and then rewrite the expression using the power rule.
Q: What is the final simplified expression for $\left(100 v^{\frac{2}{5}}\right)^{\frac{3}{2}}$ ? A: The final simplified expression is 1000 v^{\frac{3}{5}}$.

Additional Resources

For more information on the power rule of exponents, visit Math Is Fun.
For practice problems and exercises on simplifying expressions involving exponents, visit Khan Academy.

Understanding the Problem

Q&A Session

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any variables a and b and any integers m and n, (a^m)n = a^(m*n). This rule allows us to simplify expressions involving exponents by multiplying the exponents.

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

A: To apply the power rule, identify the base and the exponent inside the parentheses, and then rewrite the expression using the power rule. For example, in the expression $\left(100 v^{\frac{2}{5}}\right)^{\frac{3}{2}}$ , the base is $100 v^{\frac{2}{5}}$ and the exponent is $\frac{3}{2}$ .

Q: What is the final simplified expression for $\left(100 v^{\frac{2}{5}}\right)^{\frac{3}{2}}$ ?

A: The final simplified expression is 1000 v^{\frac{3}{5}}$. This is obtained by applying the power rule to the expression and simplifying the resulting expression.

Q: Can you provide an example of how to apply the power rule in a different expression?

A: Yes, consider the expression (2x³⁾2. Using the power rule, we can rewrite this expression as 2^2 * (x³⁾2 = 4 * x^(3*2) = 4 * x^6**.

Q: What is the difference between the power rule and the product rule of exponents?

A: The power rule states that (a^m)n = a^(m*n), while the product rule states that a^m * a^n = a^(m+n). The power rule is used to simplify expressions involving exponents, while the product rule is used to simplify expressions involving the product of variables with the same base.

Q: Can you provide a real-world example of how the power rule is used in mathematics?

A: Yes, consider the problem of calculating the area of a circle. The formula for the area of a circle is A = πr^2, where r is the radius of the circle. Using the power rule, we can rewrite this expression as A = π * (r²⁾1 = π * r^(2*1) = π * r^2**.

Conclusion

In this Q&A article, we have discussed the power rule of exponents and its application in simplifying expressions involving exponents. The power rule states that for any variables a and b and any integers m and n, (a^m)n = a^(m*n). This rule is essential in mathematics and is used in a variety of applications, including algebra and geometry.

Frequently Asked Questions

Q: What is the power rule of exponents? A: The power rule of exponents states that for any variables a and b and any integers m and n, (a^m)n = a^(m*n).
Q: How do you apply the power rule to simplify an expression? A: To apply the power rule, identify the base and the exponent inside the parentheses, and then rewrite the expression using the power rule.
Q: What is the final simplified expression for $\left(100 v^{\frac{2}{5}}\right)^{\frac{3}{2}}$ ? A: The final simplified expression is 1000 v^{\frac{3}{5}}$.

Additional Resources

For more information on the power rule of exponents, visit Math Is Fun.
For practice problems and exercises on simplifying expressions involving exponents, visit Khan Academy.