Multiply (a^3+a^2b)/(5a)*25/(3b+3a)

Solution:

Step 1: Extract $a^2$ from the numerator expression $a^3 + a^2b$.

• Step 1.1: Take $a^2$ out of $a^3$. $\frac{a^2 \cdot a + a^2 \cdot b}{5a} \cdot \frac{25}{3b + 3a}$
• Step 1.2: Take $a^2$ out of $a^2b$. $\frac{a^2 \cdot a + a^2 \cdot b}{5a} \cdot \frac{25}{3b + 3a}$
• Step 1.3: Combine the factored terms. $\frac{a^2(a + b)}{5a} \cdot \frac{25}{3b + 3a}$

Step 2: Factor out the common factor of 3 from the denominator $3b + 3a$.

• Step 2.1: Take 3 out of $3b$. $\frac{a^2(a + b)}{5a} \cdot \frac{25}{3(b + a)}$
• Step 2.2: Take 3 out of $3a$. $\frac{a^2(a + b)}{5a} \cdot \frac{25}{3(b + a)}$
• Step 2.3: Combine the factored terms. $\frac{a^2(a + b)}{5a} \cdot \frac{25}{3(b + a)}$

Step 3: Combine the expressions.

$\frac{a^2(a + b) \cdot 25}{5a \cdot 3(b + a)}$

Step 4: Simplify by canceling out the common factors of $a^2$ and $a$.

• Step 4.1: Factor $a$ from the numerator. $\frac{a(a(a + b) \cdot 25)}{5a \cdot 3(b + a)}$

• Step 4.2: Cancel the common $a$ factors.

  • Step 4.2.1: Factor $a$ from the denominator. $\frac{a(a(a + b) \cdot 25)}{a \cdot 5 \cdot 3(b + a)}$
  • Step 4.2.2: Cancel the common $a$. $\frac{\cancel{a}(a(a + b) \cdot 25)}{\cancel{a} \cdot 5 \cdot 3(b + a)}$
  • Step 4.2.3: Rewrite the expression. $\frac{a(a + b) \cdot 25}{5 \cdot 3(b + a)}$

Step 5: Cancel the common factors of 25 and 5.

• Step 5.1: Factor 5 from the numerator. $\frac{5(a(a + b) \cdot 5)}{5 \cdot 3(b + a)}$

• Step 5.2: Cancel the common 5.

  • Step 5.2.1: Cancel the common 5. $\frac{\cancel{5}(a(a + b) \cdot 5)}{\cancel{5} \cdot 3(b + a)}$
  • Step 5.2.2: Rewrite the expression. $\frac{a(a + b) \cdot 5}{3(b + a)}$

Step 6: Cancel the common factors of $a + b$ and $b + a$.

• Step 6.1: Reorder the terms. $\frac{a \cdot 5(a + b)}{3(a + b)}$
• Step 6.2: Cancel the common $a + b$. $\frac{a \cdot 5 \cancel{(a + b)}}{3 \cancel{(a + b)}}$
• Step 6.3: Rewrite the expression. $\frac{a \cdot 5}{3}$

Step 7: Rearrange the terms.

$\frac{5a}{3}$

Knowledge Notes:

• Factoring: The process of breaking down an expression into simpler components that, when multiplied together, give the original expression. This is often used to simplify algebraic expressions and solve equations.

• Common Factors: A number or algebraic term that divides two or more numbers or terms without a remainder. Identifying common factors is a key step in simplification.

• Cancellation: In fractions, if a factor appears both in the numerator and the denominator, it can be 'cancelled out' or reduced to 1, simplifying the fraction.

• Rearranging Terms: Algebraic expressions can be rewritten in different forms to simplify or solve them. This may involve factoring, expanding, or combining like terms.

• Algebraic Manipulation: The process of using algebraic techniques to rearrange and simplify expressions. This includes factoring, expanding, and using various properties of numbers (like distributive, associative, and commutative properties).

• Latex Formatting: A typesetting system that is widely used for mathematical and scientific documents due to its powerful handling of formulas and text. In this context, Latex is used to clearly present mathematical expressions.