Multiply And Simplify.$\[ 16u^4 \cdot \frac{14}{8u^2} \\]where \[$ U \neq 0 \$\].

Feb 27, 2025 by ADMIN 82 views

**Multiplying and Simplifying Algebraic Expressions**

Introduction

In algebra, multiplying and simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. In this article, we will focus on multiplying and simplifying the given expression: $16u^4 \cdot \frac{14}{8u^2}$ . This expression involves multiplying two algebraic expressions, and we will use the rules of exponents and simplification to simplify it.

Understanding the Expression

The given expression is $16u^4 \cdot \frac{14}{8u^2}$ . To simplify this expression, we need to understand the rules of exponents and how to multiply fractions. The expression consists of two parts: the numerator and the denominator. The numerator is $16u^4$ , and the denominator is $\frac{14}{8u^2}$ .

Multiplying the Numerator and Denominator

To multiply the numerator and denominator, we need to follow the rules of exponents. When multiplying two variables with the same base, we add their exponents. In this case, the base is $u$ , and the exponents are $4$ and $-2$ . Therefore, we can multiply the numerator and denominator as follows:

16u^4 \cdot \frac{14}{8u^2} = \frac{16u^4 \cdot 14}{8u^2}

Simplifying the Expression

Now that we have multiplied the numerator and denominator, we can simplify the expression by canceling out any common factors. In this case, we can cancel out the factor of $8$ in the denominator with the factor of $8$ in the numerator. We can also cancel out the factor of $u^2$ in the denominator with the factor of $u^2$ in the numerator.

\frac{16u^4 \cdot 14}{8u^2} = \frac{16 \cdot 14 \cdot u^4}{8 \cdot u^2}

Canceling Out Common Factors

Now that we have simplified the expression, we can cancel out any common factors. In this case, we can cancel out the factor of $8$ in the denominator with the factor of $8$ in the numerator. We can also cancel out the factor of $u^2$ in the denominator with the factor of $u^2$ in the numerator.

\frac{16 \cdot 14 \cdot u^4}{8 \cdot u^2} = \frac{14 \cdot u^4}{u^2}

Simplifying Further

Now that we have canceled out the common factors, we can simplify the expression further by combining the variables. In this case, we can combine the variables $u^4$ and $u^2$ by subtracting their exponents.

\frac{14 \cdot u^4}{u^2} = 14u^{4-2}

Final Answer

The final answer is $14u^2$ .

Conclusion

In this article, we have learned how to multiply and simplify the given expression: $16u^4 \cdot \frac{14}{8u^2}$ . We have used the rules of exponents and simplification to simplify the expression and arrive at the final answer: $14u^2$ . This expression involves multiplying two algebraic expressions, and we have used the rules of exponents and simplification to simplify it.

Example Problems

Here are some example problems that involve multiplying and simplifying algebraic expressions:

$12x^3 \cdot \frac{15}{4x^2}$
$9y^4 \cdot \frac{20}{6y^2}$
$16z^5 \cdot \frac{25}{8z^3}$

Solutions

Here are the solutions to the example problems:

$12x^3 \cdot \frac{15}{4x^2} = \frac{12 \cdot 15 \cdot x^3}{4 \cdot x^2} = \frac{180x^3}{4x^2} = 45x$
$9y^4 \cdot \frac{20}{6y^2} = \frac{9 \cdot 20 \cdot y^4}{6 \cdot y^2} = \frac{180y^4}{6y^2} = 30y^2$
$16z^5 \cdot \frac{25}{8z^3} = \frac{16 \cdot 25 \cdot z^5}{8 \cdot z^3} = \frac{400z^5}{8z^3} = 50z^2$

Tips and Tricks

Here are some tips and tricks for multiplying and simplifying algebraic expressions:

Always follow the order of operations (PEMDAS) when multiplying and simplifying expressions.
Use the rules of exponents to simplify expressions.
Cancel out any common factors in the numerator and denominator.
Combine variables by subtracting their exponents.

Introduction

In our previous article, we learned how to multiply and simplify algebraic expressions. In this article, we will answer some frequently asked questions about multiplying and simplifying algebraic expressions.

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

A: The order of operations when multiplying and simplifying expressions is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: To simplify an expression with multiple variables, you can use the rules of exponents to combine the variables. For example, if you have the expression $x^2y^3$ , you can simplify it by combining the variables as follows:

x^2y^3 = x^{2+3} = x^5

Q: What is the difference between multiplying and simplifying expressions?

A: Multiplying expressions involves combining two or more expressions using the multiplication operation. Simplifying expressions involves reducing an expression to its simplest form by combining like terms, canceling out common factors, and using the rules of exponents.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, you can follow these steps:

Multiply the numerator and denominator by the same value to eliminate any common factors.
Simplify the resulting expression by combining like terms and using the rules of exponents.
Cancel out any common factors in the numerator and denominator.

Q: What is the rule for multiplying variables with the same base?

A: The rule for multiplying variables with the same base is to add their exponents. For example, if you have the expression $x^2 \cdot x^3$ , you can multiply the variables as follows:

x^2 \cdot x^3 = x^{2+3} = x^5

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

A: To simplify an expression with a negative exponent, you can use the rule that $a^{-n} = \frac{1}{a^n}$ . For example, if you have the expression $x^{-2}$ , you can simplify it as follows:

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

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. A constant is a value that does not change. For example, in the expression $x^2 + 3$ , the variable is $x$ and the constant is $3$ .

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

A: To simplify an expression with multiple constants, you can combine the constants by adding or subtracting them. For example, if you have the expression $2x + 3 + 4$ , you can simplify it as follows:

2x + 3 + 4 = 2x + 7

Q: What is the rule for simplifying expressions with like terms?

A: The rule for simplifying expressions with like terms is to combine the terms by adding or subtracting their coefficients. For example, if you have the expression $2x + 3x$ , you can simplify it as follows:

2x + 3x = 5x