Problem 74 - Algebra 2 and Trigonometry

Question

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{15 x^{-2} y^{2}}{3 x y^{5}} $$

Step-by-Step Solution

Answer

The quotient as a product without a denominator is \( 5x^{-3}y^{-3} \).

Step 1: Divide the Coefficients

The fractional coefficient is \( \frac{15}{3} \). Divide the coefficients: \( 15 \div 3 = 5 \). So, the new coefficient is 5.

Step 2: Simplify the Power of x

We have \( x^{-2} \) in the numerator and \( x \) in the denominator. Subtract the exponent in the denominator from the exponent in the numerator: \( -2 - 1 = -3 \). This results in \( x^{-3} \).

Step 3: Simplify the Power of y

For \( y \, \), the fraction is \( y^2 \; / \; y^5 \). Subtract the exponent in the denominator from the exponent in the numerator: \( 2 - 5 = -3 \). This results in \( y^{-3} \).

Step 4: Write the Expression as a Product Without a Denominator

After simplifying, the expression becomes \( 5x^{-3}y^{-3} \) where there are no quotients left.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. When you see a number or variable with a small number slightly above and to the right, that small number is an exponent. It tells you how many times to multiply the base by itself. For example, in the expression \( x^2 \), the base \( x \) is multiplied by itself, producing \( x \times x \).
When working with similar bases, there are some rules to keep in mind:

• When multiplying, add the exponents: \( x^a \cdot x^b = x^{a+b} \).
• When dividing, subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).

These rules make calculations with exponents much simpler, allowing you to handle complex expressions with ease.

Division of Monomials

Monomials are algebraic expressions that contain a single term, which can be a number, a variable, or a combination of both with an exponent. When dividing one monomial by another, divide both their coefficients and variables separately.Here's how it works:
Take the coefficients first. If you have \( \frac{15}{3} \), divide to get 5. For variables like \( x^{-2} \) and \( x \), apply the exponent rule to subtract the exponents. This simplifies to \( x^{-2-1} = x^{-3} \). Do the same for any other variables such as \( y^2 / y^5 \) to get \( y^{-3} \).
This separation into coefficient and variable handling makes division of monomials manageable and straightforward.

Simplifying Expressions

Simplifying expressions means making them as concise and clear as possible, without changing their value. It's like cleaning up the math so it looks nicer and is easier to work with.
For example, in the expression \( \frac{15 x^{-2} y^{2}}{3 x y^{5}} \), you start by simplifying the coefficients, reducing \( \frac{15}{3} \) to 5. Then, handle the variables by using the exponent rules—subtract denominators' exponents from numerators'. For \( x^{-2}/x \), apply \(-2 - 1\), resulting in \( x^{-3} \); and for \( y^2/y^5 \), \(2 - 5\) becomes \( y^{-3} \). Once finished with each part, combine into \( 5x^{-3}y^{-3} \), which is now simplified.

Negative Exponents

Negative exponents can be a bit confusing at first, but they're quite simple once you understand them. A negative exponent changes the position of a base in a fraction. Specifically, \( x^{-n} = \frac{1}{x^n} \). This means a negative exponent turns a number or variable from numerator to denominator and vice versa.
For example, when you see \( x^{-3} \), it indicates \( \frac{1}{x^3} \). Therefore, in an expression like \( 5x^{-3}y^{-3} \), negative exponents highlight that both \( x^3 \) and \( y^3 \) are actually in the denominator.
Understanding negative exponents is crucial in algebra because it helps simplify expressions and write them in different forms that are often more useful in solving equations.