Problem 56 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Use only positive exponents. $$ \frac{5 x^{-3} y^{2}}{x^{5} y^{-1}} \cdot \frac{\left(2 x y^{3}\right)^{-2}}{x y} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5 / (4x^8 y^7) \)

Step 1: Distribute Exponents and Simplify

Start by distributing any exponents to terms inside of brackets, and simplifying negative exponents using the rule that \(x^{-n} = 1/x^n\). Simplify the expression: \[(5x^{-3} y^2) \cdot ((2xy^3)^{-2}) / (x^5 y^{-1} \cdot xy)\]

Step 2: Apply Exponent Laws To Simplify

Next, apply the exponents laws to further simplify the expression. This essentially involves subtracting the exponents of like terms in the numerator and the denominator. The expression becomes: \[(5 / x^2) \cdot (1 / (4x^2 y^6)) / (x^4 / y) = (5 / 4x^4 y^6) / (x^4 / y)\]

Step 3: Simplify

The expression then simplifies to: \[5 / 4x^8 y^7\]

Key Concepts

Understanding Positive Exponents

Positive exponents represent the number of times a base is multiplied by itself. For example, in the expression \( x^3 \), the base \( x \) is multiplied three times: \( x \times x \times x \). Unlike negative exponents, which indicate division or a fraction, positive exponents keep us in the realm of multiplication. In the exercise provided, to use only positive exponents, we convert negative exponents by flipping the base to the other side of the fraction. Thus, \( x^{-3} \) becomes \( 1/x^3 \) when placed in the denominator, adhering to the mandate of using only positive exponents.

Additionally, having a solid understanding of positive exponents aids in simplifying algebraic expressions since it makes the subsequent steps, such as applying exponent laws, more intuitive.

Applying Exponent Laws

Exponent laws, also known as the laws of indices, are rules that describe how to handle exponents in mathematical operations. There are several crucial laws to remember:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( a^m / a^n = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Power of a Product: \( (a \times b)^n = a^n \times b^n \)

These laws are applied in our exercise during the simplification process. For instance, the step of 'applying the exponents laws' in the given solution aligns with the Quotient of Powers law: when dividing like bases, you subtract the exponents, such as from \( x^5 / x^3 = x^{5-3} = x^2 \). These operations transform the original expression to a simpler form with positive exponents only.

Distributing Exponents in Algebraic Expressions

When an exponent is distributed across a product or fraction within brackets, we use the Power of a Product Rule: \( (ab)^n = a^n b^n \), or the Power of a Quotient Rule: \( (a/b)^n = a^n / b^n \). These rules allow us to apply the outside exponent to each term inside the brackets individually.

In our exercise, the term \( (2xy^3)^{-2} \) requires distributing the exponent \( -2 \) to both \( 2x \) and \( y^3 \), resulting in \( 2^{-2} x^{-2} y^{-6} \), which is simplified further along with other parts of the expression. Distributing exponents properly is critical in obtaining the correct simplified form. A common mistake is to apply the exponent to just one term inside the brackets, but remember, each term in the product or quotient must be raised to the power individually.