Which Law Would You Use To Simplify The Expression $\left(x^4\right$\]?A. Product Of Powers B. Power Of A Product C. Power Of A Quotient D. Power Of A Power

Introduction

When working with expressions that involve exponents, it's essential to know which laws to apply to simplify them. In this article, we'll explore the four laws of exponents and provide examples of how to use them to simplify expressions. We'll also discuss the correct answer to the question posed in the title, which law to use to simplify the expression $\left(x^4\right)^{1/2}$.

The Four Laws of Exponents

There are four laws of exponents that are commonly used to simplify expressions:

• Product of Powers: This law states that when multiplying two powers with the same base, you can add the exponents. The formula is $a^m \cdot a^n = a^{m+n}$.
• Power of a Product: This law states that when raising a product to a power, you can raise each factor to that power. The formula is $(ab)^n = a^n \cdot b^n$.
• Power of a Quotient: This law states that when raising a quotient to a power, you can raise the numerator and denominator to that power. The formula is $(\frac{a}{b})^n = \frac{a^n}{b^n}$.
• Power of a Power: This law states that when raising a power to a power, you can multiply the exponents. The formula is $(a^m)^n = a^{m \cdot n}$.

Simplifying the Expression $\left(x^4\right)^{1/2}$

Now that we've reviewed the four laws of exponents, let's apply them to simplify the expression $\left(x^4\right)^{1/2}$. To do this, we'll use the Power of a Power law, which states that when raising a power to a power, you can multiply the exponents.

Using this law, we can rewrite the expression as $x^{4 \cdot 1/2}$. Simplifying the exponent, we get $x^2$.

Therefore, the correct answer to the question posed in the title is D. Power of a Power.

Example 1: Simplifying $\left(x^3\right)^2$

Let's use the Power of a Power law to simplify the expression $\left(x^3\right)^2$. Applying the law, we get $x^{3 \cdot 2} = x^6$.

Example 2: Simplifying $\left(\frac{x^2}{y^3}\right)^4$

Let's use the Power of a Quotient law to simplify the expression $\left(\frac{x^2}{y^3}\right)^4$. Applying the law, we get $\frac{x^{2 \cdot 4}}{y^{3 \cdot 4}} = \frac{x^8}{y^{12}}$.

Example 3: Simplifying $\left(xy\right)^3$

Let's use the Power of a Product law to simplify the expression $\left(xy\right)^3$. Applying the law, we get $x^3 \cdot y^3$.

Example 4: Simplifying $\left(x^2y^3\right)^2$

Let's use the Power of a Product law to simplify the expression $\left(x^2y^3\right)^2$. Applying the law, we get $x^{2 \cdot 2} \cdot y^{3 \cdot 2} = x^4 \cdot y^6$.

Conclusion

Q: What is the difference between the Product of Powers and Power of a Product laws?

A: The Product of Powers law states that when multiplying two powers with the same base, you can add the exponents. The formula is $a^m \cdot a^n = a^{m+n}$. The Power of a Product law states that when raising a product to a power, you can raise each factor to that power. The formula is $(ab)^n = a^n \cdot b^n$.

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

A: To simplify an expression with a negative exponent, you can use the Power of a Power law. For example, to simplify $x^{-2}$, you can rewrite it as $\frac{1}{x^2}$.

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

A: Yes, you can simplify an expression with a zero exponent. When an exponent is zero, the expression is equal to 1. For example, $x^0 = 1$.

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

A: To simplify an expression with a fractional exponent, you can use the Power of a Power law. For example, to simplify $\left(x^2\right)^{1/2}$, you can rewrite it as $x^1 = x$.

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

A: Yes, you can simplify an expression with a variable in the exponent. For example, to simplify $x^{2y}$, you can rewrite it as $(x^2)^y$.

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

A: To simplify an expression with multiple exponents, you can use the Product of Powers law. For example, to simplify $x^2 \cdot x^3$, you can rewrite it as $x^{2+3} = x^5$.

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

A: Yes, you can simplify an expression with a negative base and a positive exponent. For example, to simplify $(-x)^3$, you can rewrite it as $-x^3$.

Q: How do I simplify an expression with a variable base and a variable exponent?

A: To simplify an expression with a variable base and a variable exponent, you can use the Power of a Power law. For example, to simplify $(x^y)^z$, you can rewrite it as $x^{y \cdot z}$.

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

A: Yes, you can simplify an expression with a complex number in the exponent. For example, to simplify $(x^2)^{1+i}$, you can rewrite it as $x^{2(1+i)}$.

Conclusion

In conclusion, simplifying expressions with exponents can be a challenging task, but with the right laws and techniques, it can be made easier. By understanding the four laws of exponents and how to apply them, you can simplify complex expressions and make them easier to work with. We hope this article has provided you with a better understanding of how to simplify expressions with exponents and has answered some of the most frequently asked questions in this area.