Problem 71 - Algebra 2 and Trigonometry

Question

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{36}{9 x^{-5}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 4x^5 \).

Step 1: Understand the Expression

We start with the expression \( \frac{36}{9x^{-5}} \). Our goal is to convert this into a form without a fraction.

Step 2: Simplify the Denominator

The denominator is \( 9x^{-5} \). This can be rewritten by expressing the negative exponent as a positive exponent in the numerator. Recall that \( x^{-5} = \frac{1}{x^5} \). Thus, \( 9x^{-5} = \frac{9}{x^5} \).

Step 3: Rewrite the Fraction

The expression \( \frac{36}{9x^{-5}} \) becomes \( 36 \times \frac{x^5}{9} \), using the property that dividing by a fraction is the same as multiplying by its reciprocal.

Step 4: Simplify the Expression

Now, multiply directly: \( 36 \times \frac{x^5}{9} = \frac{36}{9} \times x^5 \). Simplify \( \frac{36}{9} \) to get 4. Hence, the expression simplifies to \( 4x^5 \).

Key Concepts

Negative Exponents

Understanding negative exponents is essential in algebra, as they play a vital role in simplifying expressions. A negative exponent indicates that the base of the power should be taken as the reciprocal and raised to the corresponding positive exponent.

• For instance, if we have an expression like \(x^{-5}\), it can be rewritten as \(\frac{1}{x^5}\).
• This is because a negative exponent flips the base from the numerator to the denominator or vice versa.

By converting negative exponents into positive ones through reciprocals, calculations become straightforward, especially when dealing with quotients or products. This approach allows us to eliminate fractions by turning division into multiplication, simplifying algebraic manipulation.

Fractions

Fractions frequently appear in mathematical expressions and involve a numerator and a denominator. Simplifying fractions is key to making calculations easier. To simplify operations, it's helpful to:

• Convert complex fractions into simpler forms.
• Use properties such as multiplying by the reciprocal.

For example, a fraction like \(\frac{36}{9x^{-5}}\) can be transformed into a multiplication problem. Here, the fraction is perceived as division, which gets simplified by using the reciprocal of the denominator. This makes operations intuitive and aids in removing fractions from equations, offering a cleaner representation.

Quotients

In algebra, a quotient represents the result of dividing one quantity by another. Understanding quotients is crucial as they allow for the simplification of expressions. To effectively work with quotients:

• Identify the numerator and the denominator.
• Simplify the expression by dividing where possible.

In the expression \(\frac{36}{9x^{-5}}\), the quotient is transformed by rewriting it as a multiplication using the reciprocal of the denominator. This involves changing the division into multiplication which simplifies the computation process greatly.

Reciprocals

A reciprocal of a number is simply 1 divided by that number. It plays a key role in algebraic simplification, particularly in handling fractions. By employing reciprocals, fractions can be turned into products:

• The reciprocal of \(9x^{-5}\) is calculated as \(\frac{1}{9}x^5\).
• Using reciprocals simplifies operations, turning division into multiplication.

This concept becomes especially useful when converting problems with division into multiplication. For instance, replacing a fraction by its reciprocal allows for easier simplification and clearer results, as shown when rewriting \(\frac{36}{9x^{-5}}\) into its product form \(36 \times \frac{x^5}{9}\). By understanding and using reciprocals appropriately, complex expressions become manageable.