Problem 14 - Algebra 1

Question

Simplify the expression. $$\frac{7 x^{2}}{6 x} \cdot \frac{12 x^{2}}{2 x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(7x^{2}\).

Step 1: Rewrite Expression

Firstly, rewrite the given multiplication of fractions, putting each term next to their corresponding term, i.e., rewrite \(\frac{7 x^{2}}{6 x} \cdot \frac{12 x^{2}}{2 x}\) as \(\frac{7 x^{2} \cdot 12 x^{2}}{6 x \cdot 2 x}\)

Step 2: Simplification

Next, perform simplifications within the numerator and denominator separately. In the numerator, multiply 7 and 12 to get 84 and add the exponents to get \(x^{2+2}=x^{4}\). In the denominator, multiply 6 and 2 to get 12 and add the exponents of x to get \(x^{1+1}=x^{2}\). So our fraction now becomes \(\frac{84x^{4}}{12x^{2}}\)

Step 3: Cancel Similar Terms

Then, we cancel out the similar terms in the numerator and denominator. We have common factor 12 in the numerator and denominator, and x to the power 2, so we divide each part by their greatest common factor. This results in the fraction \(\frac{7x^{4-2}}{1}\)

Step 4: Simplify The Exponent

Now, simplify the exponent in the numerator to get \(x^{4-2}=x^{2}\). This gives the final simplified expression as \(7x^{2}\)

Key Concepts

Algebraic Fractions

Understanding algebraic fractions is crucial in simplifying complex expressions, as seen in our exercise involving \(\frac{7 x^{2}}{6 x} \cdot \frac{12 x^{2}}{2 x}\). An algebraic fraction is similar to a normal fraction, but it includes variables such as \(x\) in the numerator, denominator, or both.

In the process of simplification, the goal is to reduce the expression to its simplest form by performing operations such as factoring, canceling out common factors, and simplifying terms. When simplifying algebraic fractions, ensure to only cancel factors that appear in both the numerator and the denominator. This results in an equivalent expression that is easier to work with or evaluate.

For instance, if we have a term like \(x^{2}\) in the numerator and an \(x\) in the denominator, we can subtract the exponent in the denominator from the exponent in the numerator, reflecting the law of exponents for division: \(x^{m}/x^{n} = x^{m-n}\).

Exponent Rules

Exponent rules, often referred to as 'laws of exponents', are a set of rules that govern the operations on expressions involving powers.

When you're working with expressions that have the same base, like \(x\) in our textbook exercise, these rules can dramatically simplify your work:

• Multiplication: To multiply powers with the same base, you add the exponents, which allows us to combine \(x^{2} \times x^{2}\) into \(x^{4}\).
• Division: To divide powers with the same base, you subtract the exponents, turning \(\frac{x^{4}}{x^{2}}\) into \(x^{4-2} = x^{2}\).
• Power of a power: To raise a power to another power, multiply the exponents. For example, \( (x^{2})^{3}\) becomes \(x^{2\times3} = x^{6}\).
• Zero exponent: Any nonzero number raised to the power of zero equals 1, so \(x^{0} = 1\).

The consistent application of these rules is how we achieve the final simplification of our algebraic expression.

Numerical Operations

Numerical operations are the basic mathematical operations—addition, subtraction, multiplication, and division—applied to numbers. When dealing with algebraic expressions, these operations have to be precisely executed to avoid errors in simplification.

In simplifying the given algebraic fraction, we first performed multiplication in the numerator and denominator. This process included multiplying constants with constants and similar variables with similar variables, following the commutative law of multiplication which states that the order of numbers does not affect the product.

Moreover, division plays a key role in simplification as well. For example, when we encounter \(\frac{84}{12}\), we have to divide 84 by 12 to simplify it to 7. It's important that students pay careful attention to these numerical operations while simplifying algebraic expressions to obtain the correct final result, as any mistake with these basic operations can lead to an incorrect solution, as was necessary in reducing the fraction to \(7x^{2}\).