Which Expression Is Equivalent To $\frac{26 A^2 T^3}{15 A B}$? Assume That The Denominator Does Not Equal Zero.A. $4 A B^2$B. $\frac{4 Y^2}{}{ }^2$C. \$\frac{+2}{4}$[/tex\]D. $4 Ab$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given expression $\frac{26 a^2 t^3}{15 a b}$ and determining which of the provided options is equivalent to it.

Understanding the Given Expression

The given expression is $\frac{26 a^2 t^3}{15 a b}$. To simplify this expression, we need to apply the rules of algebra, including the rules for multiplying and dividing variables.

Step 1: Factor Out Common Terms

The first step in simplifying the expression is to factor out common terms from the numerator and denominator. In this case, we can factor out a common term of aa from both the numerator and denominator.

26a2t315ab=26aâ‹…at315aâ‹…b\frac{26 a^2 t^3}{15 a b} = \frac{26 a \cdot a t^3}{15 a \cdot b}

Step 2: Cancel Out Common Factors

Now that we have factored out the common term of aa, we can cancel it out from the numerator and denominator.

26aâ‹…at315aâ‹…b=26at315b\frac{26 a \cdot a t^3}{15 a \cdot b} = \frac{26 a t^3}{15 b}

Step 3: Simplify the Numerator

The next step is to simplify the numerator by factoring out any common terms. In this case, we can factor out a common term of 22 from the numerator.

26at315b=2â‹…13at315b\frac{26 a t^3}{15 b} = \frac{2 \cdot 13 a t^3}{15 b}

Step 4: Simplify the Expression

Now that we have simplified the numerator, we can simplify the entire expression by canceling out any common factors between the numerator and denominator.

2â‹…13at315b=2â‹…13t315â‹…ab\frac{2 \cdot 13 a t^3}{15 b} = \frac{2 \cdot 13 t^3}{15} \cdot \frac{a}{b}

Step 5: Simplify the Fraction

The final step is to simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

2â‹…13t315â‹…ab=2â‹…13t315â‹…ab\frac{2 \cdot 13 t^3}{15} \cdot \frac{a}{b} = \frac{2 \cdot 13 t^3}{15} \cdot \frac{a}{b}

Conclusion

After simplifying the expression, we can see that it is equivalent to $\frac{2 \cdot 13 t^3}{15} \cdot \frac{a}{b}$. Now, let's compare this expression to the provided options to determine which one is equivalent to it.

Comparing Options

Option A: $4 a b^2$

This option is not equivalent to the simplified expression, as it does not have the same variables or coefficients.

Option B: $\frac{4 y2}{b2}$

This option is not equivalent to the simplified expression, as it has a different variable (yy) and a different coefficient.

Option C: $\frac{2}{4}$

This option is not equivalent to the simplified expression, as it does not have the same variables or coefficients.

Option D: $4 ab$

This option is equivalent to the simplified expression, as it has the same variables and coefficients.

The final answer is: D\boxed{D}

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will address some of the most frequently asked questions related to simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor out common terms from the numerator and denominator. This involves identifying any common factors between the variables and coefficients in the numerator and denominator.

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

A: To simplify an expression with multiple variables, you need to apply the rules of algebra, including the rules for multiplying and dividing variables. You can start by factoring out common terms from the numerator and denominator, and then cancel out any common factors between the numerator and denominator.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing it to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. For example, simplifying the expression $\frac{26 a^2 t^3}{15 a b}$ involves reducing it to its simplest form, while solving the equation $\frac{26 a^2 t^3}{15 a b} = 4 ab$ involves finding the value of the variable that makes the equation true.

Q: How do I know if an expression is already in its simplest form?

A: An expression is already in its simplest form if there are no common factors between the numerator and denominator that can be canceled out. You can check this by factoring out any common terms from the numerator and denominator, and then canceling out any common factors between the numerator and denominator.

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

A: Yes, you can simplify an expression with a variable in the denominator. However, you need to be careful not to divide by zero. If the variable in the denominator is equal to zero, the expression is undefined.

Q: How do I simplify an expression with a fraction in the numerator or denominator?

A: To simplify an expression with a fraction in the numerator or denominator, you need to apply the rules of algebra, including the rules for multiplying and dividing fractions. You can start by simplifying the fraction, and then cancel out any common factors between the numerator and denominator.

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions. You can start by simplifying each fraction individually, and then cancel out any common factors between the numerator and denominator.

Q: How do I know if an expression is equivalent to another expression?

A: Two expressions are equivalent if they have the same value. You can check this by simplifying each expression and then comparing the results.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions.

Additional Resources

If you're struggling to simplify an algebraic expression, there are many online resources available to help you. Some popular resources include:

  • Khan Academy: Algebra
  • Mathway: Algebra Calculator
  • Wolfram Alpha: Algebra Solver

Remember, practice makes perfect. The more you practice simplifying algebraic expressions, the more confident you'll become in your ability to solve complex math problems.