Which Law Would You Use To Simplify The Expression $\left(\frac{p}{q}\right)^3$?A. Power Of A Power B. Power Of A Quotient C. Quotient Of Powers D. Power Of A Product

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Introduction

When dealing with expressions involving exponents, it's essential to understand the laws that govern their behavior. These laws enable us to simplify complex expressions and make them more manageable. In this article, we'll focus on the law that would be used to simplify the expression $\left(\frac{p}{q}\right)^3$.

Laws of Exponents

Before we dive into the specific law that applies to the given expression, let's briefly review the laws of exponents. There are several laws that govern how exponents interact with each other, and they are essential for simplifying expressions.

Power of a Power

The power of a power law states that when we raise a power to a power, we multiply the exponents. Mathematically, this can be represented as:

(am)n=amn(a^m)^n = a^{mn}

For example, if we have the expression $(23)4$, we can simplify it using the power of a power law:

(23)4=23⋅4=212(2^3)^4 = 2^{3 \cdot 4} = 2^{12}

Power of a Quotient

The power of a quotient law states that when we raise a quotient to a power, we raise the numerator and denominator to that power separately. Mathematically, this can be represented as:

(ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}

For example, if we have the expression $(\frac{2}{3})^4$, we can simplify it using the power of a quotient law:

(23)4=2434=1681(\frac{2}{3})^4 = \frac{2^4}{3^4} = \frac{16}{81}

Quotient of Powers

The quotient of powers law states that when we divide two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

For example, if we have the expression $\frac{25}{23}$, we can simplify it using the quotient of powers law:

2523=25−3=22=4\frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4

Power of a Product

The power of a product law states that when we raise a product to a power, we raise each factor to that power separately. Mathematically, this can be represented as:

(ab)n=anbn(ab)^n = a^n b^n

For example, if we have the expression $(2 \cdot 3)^4$, we can simplify it using the power of a product law:

(2⋅3)4=24⋅34=16⋅81=1296(2 \cdot 3)^4 = 2^4 \cdot 3^4 = 16 \cdot 81 = 1296

Simplifying the Expression

Now that we've reviewed the laws of exponents, let's apply them to the given expression $\left(\frac{p}{q}\right)^3$.

Using the power of a quotient law, we can raise the numerator and denominator to the power of 3 separately:

(pq)3=p3q3\left(\frac{p}{q}\right)^3 = \frac{p^3}{q^3}

Therefore, the correct answer is B. Power of a quotient.

Conclusion

In this article, we've explored the laws of exponents and how they can be used to simplify complex expressions. We've reviewed the power of a power, power of a quotient, quotient of powers, and power of a product laws, and applied them to the given expression $\left(\frac{p}{q}\right)^3$. By understanding these laws, we can simplify expressions and make them more manageable.

Frequently Asked Questions

  • What is the power of a quotient law? The power of a quotient law states that when we raise a quotient to a power, we raise the numerator and denominator to that power separately.
  • How do we simplify the expression $\left(\frac{p}{q}\right)^3$? We can simplify the expression by raising the numerator and denominator to the power of 3 separately, using the power of a quotient law.
  • What is the correct answer? The correct answer is B. Power of a quotient.

References

  • [1] "Laws of Exponents" by Math Open Reference
  • [2] "Exponents and Powers" by Khan Academy
  • [3] "Simplifying Expressions with Exponents" by Purplemath

Introduction

In our previous article, we explored the laws of exponents and how they can be used to simplify complex expressions. We reviewed the power of a power, power of a quotient, quotient of powers, and power of a product laws, and applied them to the expression $\left(\frac{p}{q}\right)^3$. In this article, we'll answer some frequently asked questions about simplifying expressions with exponents.

Q&A

Q: What is the power of a power law?

A: The power of a power law states that when we raise a power to a power, we multiply the exponents. Mathematically, this can be represented as:

(am)n=amn(a^m)^n = a^{mn}

For example, if we have the expression $(23)4$, we can simplify it using the power of a power law:

(23)4=23⋅4=212(2^3)^4 = 2^{3 \cdot 4} = 2^{12}

Q: How do I simplify the expression $\left(\frac{p}{q}\right)^3$?

A: We can simplify the expression by raising the numerator and denominator to the power of 3 separately, using the power of a quotient law:

(pq)3=p3q3\left(\frac{p}{q}\right)^3 = \frac{p^3}{q^3}

Q: What is the difference between the power of a quotient and the quotient of powers laws?

A: The power of a quotient law states that when we raise a quotient to a power, we raise the numerator and denominator to that power separately. Mathematically, this can be represented as:

(ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}

On the other hand, the quotient of powers law states that when we divide two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

For example, if we have the expression $\frac{25}{23}$, we can simplify it using the quotient of powers law:

2523=25−3=22=4\frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4

Q: Can I use the power of a product law to simplify the expression $\left(\frac{p}{q}\right)^3$?

A: No, the power of a product law states that when we raise a product to a power, we raise each factor to that power separately. Mathematically, this can be represented as:

(ab)n=anbn(ab)^n = a^n b^n

This law does not apply to the expression $\left(\frac{p}{q}\right)^3$, which is a quotient, not a product.

Q: How do I simplify the expression $(2 \cdot 3)^4$?

A: We can simplify the expression by raising each factor to the power of 4 separately, using the power of a product law:

(2⋅3)4=24⋅34=16⋅81=1296(2 \cdot 3)^4 = 2^4 \cdot 3^4 = 16 \cdot 81 = 1296

Conclusion

In this article, we've answered some frequently asked questions about simplifying expressions with exponents. We've reviewed the power of a power, power of a quotient, quotient of powers, and power of a product laws, and provided examples of how to apply them to simplify complex expressions.

Frequently Asked Questions

  • What is the power of a power law?
  • How do I simplify the expression $\left(\frac{p}{q}\right)^3$?
  • What is the difference between the power of a quotient and the quotient of powers laws?
  • Can I use the power of a product law to simplify the expression $\left(\frac{p}{q}\right)^3$?
  • How do I simplify the expression $(2 \cdot 3)^4$?

References

  • [1] "Laws of Exponents" by Math Open Reference
  • [2] "Exponents and Powers" by Khan Academy
  • [3] "Simplifying Expressions with Exponents" by Purplemath