Which Expression Is Equivalent To $\frac{76 A^2 B^3}{19 A B}$? Assume That The Denominator Does Not Equal Zero.A. $4 A B^2$B. $\frac{4 B^2}{a}$C. $\frac{a B^2}{4}$D. $4 A \hbar$

Feb 28, 2025 by ADMIN 178 views

$Which Expression is Equivalent to $\frac{76 a^2 b^3}{19 a b}$?$

Simplifying Algebraic Expressions

When simplifying algebraic expressions, we need to reduce them to their simplest form by canceling out common factors in the numerator and denominator. This process is essential in mathematics, as it helps us to solve equations and inequalities more efficiently.

Understanding the Problem

The given expression is $\frac{76 a^2 b^3}{19 a b}$ . To simplify this expression, we need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest factor that divides both numbers without leaving a remainder.

Finding the Greatest Common Factor

To find the GCF of 76 and 19, we can list the factors of each number. The factors of 76 are 1, 2, 4, 19, 38, and 76. The factors of 19 are 1 and 19. The greatest common factor of 76 and 19 is 19.

Canceling Out Common Factors

Now that we have found the GCF, we can cancel out the common factors in the numerator and denominator. The expression becomes $\frac{76 a^2 b^3}{19 a b} = \frac{4 a b^2}{1}$ .

Simplifying the Expression

The expression $\frac{4 a b^2}{1}$ can be simplified further by canceling out the 1 in the denominator. This leaves us with the simplified expression $4 a b^2$ .

Conclusion

The simplified expression is $4 a b^2$ . This is the equivalent expression to $\frac{76 a^2 b^3}{19 a b}$ .

Discussion

The correct answer is A. $4 a b^2$ . This is because the expression $\frac{76 a^2 b^3}{19 a b}$ can be simplified to $4 a b^2$ by canceling out the common factors in the numerator and denominator.

Common Mistakes

One common mistake when simplifying algebraic expressions is to forget to cancel out common factors. This can lead to incorrect solutions and equations. It is essential to carefully examine the numerator and denominator to ensure that all common factors are canceled out.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, as it helps us to solve equations and inequalities more efficiently. This skill is also essential in real-world applications, such as physics, engineering, and computer science.

Final Answer

The final answer is A. $4 a b^2$ .

Q: What is the greatest common factor (GCF) of two numbers?

A: The greatest common factor (GCF) of two numbers is the largest factor that divides both numbers without leaving a remainder. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I find the GCF of two numbers?

A: To find the GCF of two numbers, you can list the factors of each number and find the largest factor that they have in common. Alternatively, you can use the prime factorization method, which involves breaking down each number into its prime factors and then finding the product of the common prime factors.

Q: What is the difference between a factor and a multiple?

A: A factor is a number that divides another number without leaving a remainder. For example, 3 is a factor of 6, because 3 divides 6 without leaving a remainder. A multiple, on the other hand, is a number that is the product of a given number and an integer. For example, 6 is a multiple of 3, because 6 is the product of 3 and 2.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to reduce it to its simplest form by canceling out common factors in the numerator and denominator. This involves finding the greatest common factor (GCF) of the numerator and denominator and then canceling out the GCF.

Q: What is the order of operations in simplifying algebraic expressions?

A: The order of operations in simplifying algebraic expressions is as follows:

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 handle negative exponents in algebraic expressions?

A: When simplifying algebraic expressions with negative exponents, you can move the negative exponent to the other side of the fraction by changing its sign. For example, if you have the expression $\frac{1}{x^{-2}}$ , you can rewrite it as $x^2$ .

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. An irrational expression, on the other hand, is an expression that cannot be simplified to a rational number.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to reduce it to its simplest form by canceling out common factors in the numerator and denominator. This involves finding the greatest common factor (GCF) of the numerator and denominator and then canceling out the GCF.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is A. $4 a b^2$ . This is because the expression $\frac{76 a^2 b^3}{19 a b}$ can be simplified to $4 a b^2$ by canceling out the common factors in the numerator and denominator.