Express The Following Fraction In Simplest Form, Only Using Positive Exponents. 4 X ( 3 X 3 ) 3 \frac{4 X}{(3 X^3)^3} ( 3 X 3 ) 3 4 X

Mar 8, 2025 by ADMIN 136 views

**Simplifying Fractions with Exponents: A Step-by-Step Guide**

Introduction

In mathematics, simplifying fractions with exponents is an essential skill that helps us express complex expressions in their simplest form. In this article, we will focus on simplifying the fraction $\frac{4 x}{(3 x^3)^3}$ using only positive exponents.

Understanding Exponents

Before we dive into simplifying the fraction, let's quickly review what exponents are. An exponent is a small number that is raised to a power, indicating how many times the base number should be multiplied by itself. For example, $x^3$ means $x$ multiplied by itself three times, or $x \times x \times x$ .

Simplifying the Fraction

Now that we have a basic understanding of exponents, let's simplify the fraction $\frac{4 x}{(3 x^3)^3}$ . To simplify this fraction, we need to apply the rules of exponents.

Step 1: Apply the Power Rule

The power rule states that when we raise a power to another power, we multiply the exponents. In this case, we have $(3 x^3)^3$ , which means we need to multiply the exponent $3$ by $3$ .

(3 x^3)^3 = 3^3 \times (x^3)^3

Using the power rule, we can simplify the expression further:

3^3 \times (x^3)^3 = 27 \times x^{3 \times 3}

Step 2: Simplify the Exponent

Now that we have $x^{3 \times 3}$ , we can simplify the exponent by multiplying $3$ by $3$ .

x^{3 \times 3} = x^{9}

So, the fraction becomes:

\frac{4 x}{(3 x^3)^3} = \frac{4 x}{27 x^9}

Step 3: Cancel Out Common Factors

Now that we have the fraction in the form $\frac{4 x}{27 x^9}$ , we can cancel out common factors. In this case, we can cancel out one $x$ from the numerator and the denominator.

\frac{4 x}{27 x^9} = \frac{4}{27 x^8}

Step 4: Simplify the Fraction

Finally, we can simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $4$ and $27$ is $1$ , so we cannot simplify the fraction further.

\frac{4}{27 x^8} = \frac{4}{27 x^8}

Conclusion

In this article, we simplified the fraction $\frac{4 x}{(3 x^3)^3}$ using only positive exponents. We applied the power rule, simplified the exponent, canceled out common factors, and finally simplified the fraction. The simplified fraction is $\frac{4}{27 x^8}$ .

Tips and Tricks

When simplifying fractions with exponents, always apply the power rule first.
Simplify the exponent by multiplying the exponents.
Cancel out common factors to simplify the fraction.
Finally, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD).

Practice Problems

Try simplifying the following fractions using only positive exponents:

$\frac{2 x^2}{(4 x^4)^2}$
$\frac{3 x^3}{(2 x^2)^3}$
$\frac{5 x^5}{(3 x^3)^5}$

Answer Key

$\frac{2 x^2}{(4 x^4)^2} = \frac{2 x^2}{16 x^8} = \frac{1}{8 x^6}$
$\frac{3 x^3}{(2 x^2)^3} = \frac{3 x^3}{8 x^6} = \frac{3}{8 x^3}$
$\frac{5 x^5}{(3 x^3)^5} = \frac{5 x^5}{243 x^{15}} = \frac{5}{243 x^{10}}$
Simplifying Fractions with Exponents: Q&A =============================================

Introduction

In our previous article, we simplified the fraction $\frac{4 x}{(3 x^3)^3}$ using only positive exponents. In this article, we will answer some frequently asked questions (FAQs) about simplifying fractions with exponents.

Q: What is the power rule in simplifying fractions with exponents?

A: The power rule states that when we raise a power to another power, we multiply the exponents. For example, $(x^2)^3 = x^{2 \times 3} = x^6$ .

Q: How do I simplify the exponent in a fraction with exponents?

A: To simplify the exponent, multiply the exponents. For example, $x^{3 \times 3} = x^9$ .

Q: Can I cancel out common factors in a fraction with exponents?

A: Yes, you can cancel out common factors in a fraction with exponents. For example, $\frac{4 x}{27 x^9} = \frac{4}{27 x^8}$ .

Q: What is the greatest common divisor (GCD) in simplifying fractions with exponents?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction. For example, the GCD of $4$ and $27$ is $1$ , so we cannot simplify the fraction $\frac{4}{27 x^8}$ further.

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, change the sign of the exponent and simplify the fraction. For example, $\frac{1}{x^{-3}} = x^3$ .

Q: Can I simplify a fraction with a zero exponent?

A: Yes, you can simplify a fraction with a zero exponent. For example, $x^0 = 1$ .

Q: What is the difference between a fraction with exponents and a fraction without exponents?

A: A fraction with exponents has a base number raised to a power, while a fraction without exponents does not have a base number raised to a power. For example, $\frac{4 x}{27 x^8}$ is a fraction with exponents, while $\frac{4}{27}$ is a fraction without exponents.

Q: Can I simplify a fraction with a variable in the denominator?

A: Yes, you can simplify a fraction with a variable in the denominator. For example, $\frac{4 x}{27 x^8} = \frac{4}{27 x^7}$ .

Q: How do I simplify a fraction with a complex number in the denominator?

A: To simplify a fraction with a complex number in the denominator, multiply the numerator and the denominator by the conjugate of the denominator. For example, $\frac{1}{1 + i} = \frac{1}{1 + i} \times \frac{1 - i}{1 - i} = \frac{1 - i}{2}$ .