Simplify The Expression:${ \frac{8 P^3}{3 Q^2} \times \frac{9 Q 3}{p 4} \div \frac{16 P^2}{q} }$

Introduction

In this article, we will simplify the given expression using the rules of exponents and fractions. The expression involves multiplication and division of fractions, which can be simplified using the properties of exponents and fractions.

The Expression

The given expression is:

$$\frac{8 p^3}{3 q^2} \times \frac{9 q^3}{p^4} \div \frac{16 p^2}{q}$$

Step 1: Multiply the Numerators and Denominators

To simplify the expression, we will first multiply the numerators and denominators separately.

• Multiply the numerators: $8 p^3 \times 9 q^3 \times q$
• Multiply the denominators: $3 q^2 \times p^4 \times 16 p^2$

Step 2: Simplify the Numerators and Denominators

Now, we will simplify the numerators and denominators separately.

• Simplify the numerators: $8 p^3 \times 9 q^3 \times q = 72 p^3 q^4$
• Simplify the denominators: $3 q^2 \times p^4 \times 16 p^2 = 48 p^6 q^2$

Step 3: Divide the Numerators and Denominators

Now, we will divide the numerators and denominators separately.

• Divide the numerators: $\frac{72 p^3 q^4}{1}$
• Divide the denominators: $\frac{48 p^6 q^2}{1}$

Step 4: Simplify the Expression

Now, we will simplify the expression by dividing the numerators and denominators.

$$\frac{72 p^3 q^4}{48 p^6 q^2}$$

Step 5: Cancel Out Common Factors

We can cancel out common factors in the numerator and denominator.

• Cancel out $p^3$ in the numerator and denominator: $\frac{72}{48} \times \frac{q^4}{p^3}$
• Cancel out $q^2$ in the numerator and denominator: $\frac{72}{48} \times \frac{q^2}{p^3}$

Step 6: Simplify the Expression

Now, we will simplify the expression by canceling out common factors.

$$\frac{72}{48} \times \frac{q^2}{p^3}$$

Step 7: Simplify the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

• Simplify the fraction: $\frac{72}{48} = \frac{3}{2}$

Step 8: Simplify the Expression

Now, we will simplify the expression by simplifying the fraction.

$$\frac{3}{2} \times \frac{q^2}{p^3}$$

Step 9: Multiply the Numerators and Denominators

We can multiply the numerators and denominators separately.

• Multiply the numerators: $3 \times q^2 = 3q^2$
• Multiply the denominators: $2 \times p^3 = 2p^3$

Step 10: Simplify the Expression

Now, we will simplify the expression by multiplying the numerators and denominators.

$$\frac{3q^2}{2p^3}$$

Conclusion

In this article, we simplified the given expression using the rules of exponents and fractions. We multiplied the numerators and denominators, canceled out common factors, and simplified the fraction to arrive at the final expression.

Final Answer

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Introduction

In our previous article, we simplified the given expression using the rules of exponents and fractions. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q: What are the rules of exponents?

A: The rules of exponents are used to simplify expressions that involve exponents. The main rules of exponents are:

• Product Rule: $a^m \times a^n = a^{m+n}$
• Power Rule: $(a^m)^n = a^{mn}$
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

Q: What is the difference between multiplication and division of fractions?

A: When multiplying fractions, you multiply the numerators and denominators separately. When dividing fractions, you invert the second fraction and multiply.

Q: How do I cancel out common factors in a fraction?

A: To cancel out common factors in a fraction, you need to identify the common factors in the numerator and denominator and cancel them out. This will simplify the fraction.

Q: What is the order of operations when simplifying expressions?

A: The order of operations when simplifying expressions is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

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

A: To simplify an expression with multiple fractions, you need to follow the order of operations and simplify each fraction separately. Then, you can combine the simplified fractions.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions are:

• Not following the order of operations: Make sure to follow the order of operations when simplifying expressions.
• Not canceling out common factors: Make sure to cancel out common factors in the numerator and denominator.
• Not simplifying fractions: Make sure to simplify fractions by dividing the numerator and denominator by their GCD.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions. We covered the rules of exponents, simplifying fractions, and the order of operations. We also discussed common mistakes to avoid when simplifying expressions.

Final Tips

• Practice, practice, practice: The more you practice simplifying expressions, the better you will become.
• Use online resources: There are many online resources available that can help you simplify expressions, such as calculators and online math tools.
• Ask for help: If you are struggling to simplify an expression, don't be afraid to ask for help.