Applying Exponent Rules for Fractions

Oct 13, 2025 by miodeya.com 99 views

Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. The expression ${$ \left(\frac{4a}{b}\right)^2$}$ involves squaring a fraction. To simplify this expression, we need to apply the exponent rules for fractions.

Applying Exponent Rules for Fractions

The exponent rule for fractions states that when we raise a fraction to a power, we raise both the numerator and the denominator to that power. In this case, we have ${$ \left(\frac{4a}{b}\right)^2$}$. To simplify this expression, we need to square both the numerator and the denominator.

Squaring the Numerator and Denominator

To square the numerator, we need to multiply it by itself: $ $4a \cdot 4a = 16a^2\$}$ . To square the denominator, we need to multiply it by itself ${$ b \cdot b = b^2$$.

Simplifying the Expression

Now that we have squared the numerator and denominator, we can simplify the expression by dividing the squared numerator by the squared denominator: ${$ \frac{16a^2}{b2}$}$.

Comparing with the Options

Let's compare our simplified expression with the options provided:

A) ${ $16a^2\$}$ is incorrect because it doesn't account for the squared denominator.
B) ${$ \frac{16a^2}{b2}$}$ matches our simplified expression.
C) ${$ \frac{4b^2}{a2}$}$ is incorrect because it has the wrong numerator and denominator.
D) ${$ \frac{4a^2}{b2}$}$ is incorrect because it doesn't account for the squared numerator.

Conclusion

The simplified expression for ${$ \left(\frac{4a}{b}\right)^2$}$ is ${$ \frac{16a^2}{b2}$}$. This is the correct answer among the options provided.

Frequently Asked Questions

What is the exponent rule for fractions? The exponent rule for fractions states that when we raise a fraction to a power, we raise both the numerator and the denominator to that power.
How do we square a fraction? To square a fraction, we square both the numerator and the denominator.
What is the simplified expression for ${$ \left(\frac{4a}{b}\right)^2$}$? The simplified expression for ${$ \left(\frac{4a}{b}\right)^2$}$ is ${$ \frac{16a^2}{b2}$}$.

Additional Resources

Exponent rules for fractions
Squaring fractions
Simplifying expressions with exponents

Final Answer

The final answer is B) ${$ \frac{16a^2}{b2}$}$.

Frequently Asked Questions

Q: What is the exponent rule for fractions?

A: The exponent rule for fractions states that when we raise a fraction to a power, we raise both the numerator and the denominator to that power.

Q: How do we square a fraction?

A: To square a fraction, we square both the numerator and the denominator. For example, to square the fraction ${$ \frac{4a}{b}$}$, we would multiply the numerator by itself and the denominator by itself.

Q: What is the simplified expression for ${$ \left(\frac{4a}{b}\right)^2$}$?

A: The simplified expression for ${$ \left(\frac{4a}{b}\right)^2$}$ is ${$ \frac{16a^2}{b2}$}$.

Q: How do we simplify expressions with exponents?

A: To simplify expressions with exponents, we need to apply the exponent rules for fractions. This involves squaring both the numerator and the denominator, and then simplifying the resulting expression.

Q: What is the difference between ${$ \left(\frac{4a}{b}\right)^2$}$ and ${$ \left(\frac{4a}{b}\right)^3$}$?

A: The main difference between ${$ \left(\frac{4a}{b}\right)^2$}$ and ${$ \left(\frac{4a}{b}\right)^3$}$ is the exponent. In the first expression, we are squaring the fraction, while in the second expression, we are cubing the fraction.

Q: How do we simplify ${$ \left(\frac{4a}{b}\right)^3$}$?

A: To simplify ${$ \left(\frac{4a}{b}\right)^3$}$, we need to cube both the numerator and the denominator. This involves multiplying the numerator by itself three times and the denominator by itself three times.

Q: What is the simplified expression for ${$ \left(\frac{4a}{b}\right)^3$}$?

A: The simplified expression for ${$ \left(\frac{4a}{b}\right)^3$}$ is ${$ \frac{64a^3}{b3}$}$.

Q: How do we simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we need to apply the rule that states ${$ a^{-n} = \frac{1}{a^n}$}$. This involves flipping the fraction and changing the sign of the exponent.

Q: What is the simplified expression for ${$ \left(\frac{4a}{b}\right)^{-2}$}$?

A: The simplified expression for ${$ \left(\frac{4a}{b}\right)^{-2}$}$ is ${$ \frac{b^2}{16a2}$}$.

Additional Resources

Exponent rules for fractions
Squaring fractions
Simplifying expressions with exponents
Negative exponents

Final Answer

The final answer is B) ${$ \frac{16a^2}{b2}$}$.