Simplify The Expression And Eliminate Any Negative Exponent(s). Assume That All Letters Denote Positive Numbers.(a) $\sqrt{x^3}$\square$(b) $\sqrt[4]{x^5}$\square$

Simplify the Expression and Eliminate Any Negative Exponent(s)

In mathematics, simplifying expressions and eliminating negative exponents are essential skills for solving various mathematical problems. A negative exponent is a fraction with a negative power, and it can be simplified by taking the reciprocal of the base and changing the sign of the exponent. In this article, we will simplify two expressions containing square roots and eliminate any negative exponents.

Simplifying Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. When simplifying square roots, we can use the following properties:

• The square root of a product is the product of the square roots.
• The square root of a quotient is the quotient of the square roots.

Using these properties, we can simplify the first expression:

(a) $\sqrt{x^3}$

To simplify this expression, we can use the property that the square root of a product is the product of the square roots. We can rewrite $x^3$ as $x \cdot x \cdot x$, and then take the square root of each factor:

$$\sqrt{x^3} = \sqrt{x \cdot x \cdot x} = \sqrt{x} \cdot \sqrt{x} \cdot \sqrt{x} = x\sqrt{x}$$

Therefore, the simplified expression is $x\sqrt{x}$.

(b) $\sqrt[4]{x^5}$

To simplify this expression, we can use the property that the fourth root of a product is the product of the fourth roots. We can rewrite $x^5$ as $x \cdot x \cdot x \cdot x \cdot x$, and then take the fourth root of each factor:

$$\sqrt[4]{x^5} = \sqrt[4]{x \cdot x \cdot x \cdot x \cdot x} = \sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} \cdot \sqrt[4]{x} = x\sqrt[3]{x}$$

Therefore, the simplified expression is $x\sqrt[3]{x}$.

Eliminating Negative Exponents

A negative exponent is a fraction with a negative power. To eliminate a negative exponent, we can take the reciprocal of the base and change the sign of the exponent. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$.

Using this property, we can eliminate the negative exponents in the simplified expressions:

(a) $x\sqrt{x}$

Since the exponent of $x$ is 1/2, we can rewrite it as $x^{1/2}$. To eliminate the negative exponent, we can take the reciprocal of the base and change the sign of the exponent:

$$x^{1/2} = \frac{1}{x^{-1/2}}$$

Therefore, the expression with no negative exponents is $\frac{1}{\sqrt{x}}$.

(b) $x\sqrt[3]{x}$

Since the exponent of $x$ is 1/3, we can rewrite it as $x^{1/3}$. To eliminate the negative exponent, we can take the reciprocal of the base and change the sign of the exponent:

$$x^{1/3} = \frac{1}{x^{-1/3}}$$

Therefore, the expression with no negative exponents is $\frac{1}{\sqrt[3]{x}}$.

In this article, we simplified two expressions containing square roots and eliminated any negative exponents. We used the properties of square roots and negative exponents to simplify the expressions and eliminate the negative exponents. The simplified expressions are $x\sqrt{x}$ and $x\sqrt[3]{x}$, and the expressions with no negative exponents are $\frac{1}{\sqrt{x}}$ and $\frac{1}{\sqrt[3]{x}}$. These results demonstrate the importance of simplifying expressions and eliminating negative exponents in mathematics.

The final answer is:

• (a) $\frac{1}{\sqrt{x}}$
• (b) $\frac{1}{\sqrt[3]{x}}$
  Simplify the Expression and Eliminate Any Negative Exponent(s) - Q&A

In our previous article, we simplified two expressions containing square roots and eliminated any negative exponents. In this article, we will answer some frequently asked questions related to simplifying expressions and eliminating negative exponents.

Q: What is a negative exponent?

A: A negative exponent is a fraction with a negative power. For example, $x^{-2}$ is a negative exponent because the power is -2.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you can take the reciprocal of the base and change the sign of the exponent. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$.

Q: What is the difference between a square root and a fourth root?

A: A square root is the inverse operation of squaring a number, while a fourth root is the inverse operation of raising a number to the power of 4. For example, $\sqrt{x}$ is the square root of $x$, while $\sqrt[4]{x}$ is the fourth root of $x$.

Q: How do I simplify a square root?

A: To simplify a square root, you can use the property that the square root of a product is the product of the square roots. For example, $\sqrt{x \cdot y}$ can be rewritten as $\sqrt{x} \cdot \sqrt{y}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is a power that is greater than or equal to 0, while a negative exponent is a power that is less than 0. For example, $x^2$ is a positive exponent, while $x^{-2}$ is a negative exponent.

Q: How do I eliminate a negative exponent?

A: To eliminate a negative exponent, you can take the reciprocal of the base and change the sign of the exponent. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$.

Q: What is the final answer for the expression $\sqrt{x^3}$?

A: The final answer for the expression $\sqrt{x^3}$ is $x\sqrt{x}$.

Q: What is the final answer for the expression $\sqrt[4]{x^5}$?

A: The final answer for the expression $\sqrt[4]{x^5}$ is $x\sqrt[3]{x}$.

Q: What is the final answer for the expression $x\sqrt{x}$ with no negative exponents?

A: The final answer for the expression $x\sqrt{x}$ with no negative exponents is $\frac{1}{\sqrt{x}}$.

Q: What is the final answer for the expression $x\sqrt[3]{x}$ with no negative exponents?

A: The final answer for the expression $x\sqrt[3]{x}$ with no negative exponents is $\frac{1}{\sqrt[3]{x}}$.

In this article, we answered some frequently asked questions related to simplifying expressions and eliminating negative exponents. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.

The final answer is:

• (a) $\frac{1}{\sqrt{x}}$
• (b) $\frac{1}{\sqrt[3]{x}}$