Problem 69 - Algebra 2 and Trigonometry

Question

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{3}{x^{4}} $$

Step-by-Step Solution

Answer

\( \frac{3}{x^4} \) is equivalent to \( 3 \times x^{-4} \).

Step 1: Understanding the Problem

We need to rewrite the given quotient \( \frac{3}{x^4} \) as a product without a denominator. This means expressing it as a multiplication instead of a division.

Step 2: Convert Division to Multiplication

To eliminate the division, we use the rule that \( \frac{a}{b} = a \times b^{-1} \). This implies that \( \frac{3}{x^4} = 3 \times x^{-4} \), thus writing the expression as a product.

Key Concepts

Quotient to Product Conversion.

Quotients often need to be converted into products, especially in algebraic manipulation. This concept involves transforming an expression that includes division into an equivalent expression that uses multiplication. Let's break it down simply: when you have a division, like

• \( \frac{a}{b} \)

you can change it into a multiplication by flipping the divisor to its reciprocal. Thus, the expression becomes

• \( a \times \frac{1}{b} \).

In algebra, this is especially useful since it simplifies the expression and makes further operations easier to handle. It transforms complex division into straightforward multiplication, which is a foundational skill for tackling more intricate algebraic problems.

Negative Exponents Explained.

Understanding negative exponents is vital in algebra. They might seem tricky at first, but they actually simplify many problems. A negative exponent tells you how many times to divide by the base. For example:

• \( b^{-n} \) is the same as \( \frac{1}{b^n} \).

This means that instead of multiplying the base, you divide the base according to the exponent value. So, if you have

• \( x^{-4} \)

this becomes equivalent to

• \( \frac{1}{x^4} \),

which is exactly what we use in the initial example of converting \( \frac{3}{x^4} \) to a multiplication form like \( 3 \times x^{-4} \). Negative exponents are incredibly handy for simplifying expressions and solving algebraic equations.

Division in Algebra Simplified.

In algebra, division is one of the key operations, but it often requires careful handling, especially when variables are involved. Division in algebra deals with ratios and relationships between numbers and/or variables. Consider a simple division example

• \( \frac{a}{b} \)

where \( b \) is not zero. In algebra, this is where we see insertion of variables making expressions more complex. However, conversions and identities come into play effectively, such as reversing division into multiplication (using reciprocal), or simplifying through cancellation of common terms.
Variables add versatility but require a solid understanding of properties like associative, commutative, and identity properties of operations. Becoming comfortable with division in algebra is crucial as it sets the stage for dealing with fractions, rational expressions, and advanced algebra topics efficiently.