Problem 69 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{-3 a^{2}}{4 b} \cdot \frac{-8 b^{3}}{15 a} $$

Step-by-Step Solution

Answer

Answer: \(\frac{2}{5}ab^{2}\)

Step 1: Identify variables and powers

The given expression consists of two fractions, each containing variables 'a' and 'b' raised to some power: $$ \frac{-3 a^{2}}{4 b} \cdot \frac{-8 b^{3}}{15 a} $$ Here, in the first fraction, \(a\) has a power of 2, and \(b\) has a power of 1, while in the second fraction, \(a\) has a power of 1, and \(b\) has a power of 3.

Step 2: Multiply the numerators and denominators

Now, let's multiply the numerators together and the denominators together: Numerator: \((-3 a^{2}) \cdot (-8 b^{3})\) Denominator: \((4 b) \cdot (15 a)\)

Step 3: Simplify the products

Let's simplify the products: Numerator: \((-3 \cdot -8) \cdot (a^{2} \cdot b^{3}) = 24a^{2}b^{3}\) Denominator: \((4 \cdot 15) \cdot (a \cdot b) = 60ab\)

Step 4: Write the product as a fraction

Now, let's write the product as a fraction: $$ \frac{24a^{2}b^{3}}{60ab} $$

Step 5: Simplify the fraction

To simplify the fraction, we can cancel out any common factors between the numerator and denominator. Let's start by canceling out the common factors of 12: $$ \frac{12 \cdot 2 \cdot a^{2}b^{3}}{12 \cdot 5 \cdot ab} $$ Now, let's cancel out the common factors of \(a\) and \(b\). Since the powers of \(a\) and \(b\) are smaller in the denominator, cancel out these lower powers: $$ \frac{2}{5} \cdot \frac{a^{(2 - 1)}b^{(3 - 1)}}{1} = \frac{2}{5}ab^{2} $$ So, the final simplified expression is: $$ \frac{2}{5}ab^{2} $$

Key Concepts

Multiplying Fractions

When faced with an algebra problem that involves multiplying fractions, the key is to perform the operation step by step. Start by multiplying the numerators (the top numbers) of the fractions together, and then multiply the denominators (the bottom numbers). It's similar to dealing with regular fractions, except that with algebraic fractions, you also have variables along with coefficients (the numerical factors).

For example, take the product of \( \frac{-3 a^{2}}{4 b} \cdot \frac{-8 b^{3}}{15 a} \). We multiply the coefficients, -3 and -8, to get 24. Then we multiply the variable parts. This entails using the rule of exponents, which states that when you multiply powers with the same base, you add their exponents. So, \( a^{2} \cdot a^{1} = a^{2+1} = a^{3} \) and \( b^{1} \cdot b^{3} = b^{1+3} = b^{4} \). As a result, the product of the numerators is \( 24a^{3}b^{4} \).

This process simplifies complex expressions and helps you obtain a product that is easier to work with in the subsequent steps of simplification.

Variables and Powers in Algebra

Understanding variables and powers is crucial when working with algebraic expressions. A variable, usually represented by a letter, stands for an unknown value. In the expression \( \frac{-3 a^{2}}{4 b} \cdot \frac{-8 b^{3}}{15 a} \), 'a' and 'b' are variables. Powers, also known as exponents, indicate how many times a number or variable is multiplied by itself. For example, \( a^{2} \) means \( a \) is multiplied by itself once, resulting in \( aa \).

When multiplying variables with exponents, if the variables are alike, you add the exponents per the laws of exponents. But if you're dividing, you subtract the exponents of like variables. This is visible in our example where \( a^{2} \) is divided by \( a \) resulting in \( a^{2-1} = a^{1} = a \) and \( b^{3} \) is divided by \( b \) leading to \( b^{3-1} = b^{2} \). It is important to remember these rules when working with variables and powers to avoid any mistakes during simplification.

Simplifying Algebraic Expressions

The goal of simplifying algebraic expressions is to make them as straightforward as possible. Simplification may involve reducing fractions, factoring, canceling out common factors, or using the distributive property to combine like terms.

In our example, once we have the fraction \( \frac{24a^{3}b^{4}}{60ab} \), we notice both the numerator and denominator have common numerical factors and variables. We divide both by the greatest common factor, which is 12 in this case, reducing our fraction to \( \frac{2a^{3}b^{4}}{5ab} \). Then, we simplify further by subtracting the exponents of like variables in the numerator and denominator, ultimately simplifying it to \( \frac{2}{5}ab^{2} \).

Exercise Improvement Advice

When practicing these types of problems, always look for the greatest common factor first. It will make the simplification process smoother. Also, ensure you understand the exponent rules for multiplication and division to avoid confusion. Practice often with different variables and exponents to gain confidence and proficiency in simplifying algebraic fractions.