Problem 33 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ x^{3} y^{2}(x-3)^{-7} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: \(x^{3} y^{2}(x-3)^{-7}\) Answer: The rewritten expression with only positive exponents is \(\frac{x^{3}y^{2}}{(x-3)^{7}}\).

Step 1: Identify the term with a negative exponent

In the given expression, \((x-3)^{-7}\) is the term with a negative exponent.

Step 2: Rewrite the term with a positive exponent

To rewrite \((x-3)^{-7}\) with a positive exponent, we need to take its reciprocal, which can be done by moving \((x-3)^{-7}\) from the numerator to the denominator and changing the negative exponent to a positive exponent. Hence, we get: $$ (x-3)^{-7} = \frac{1}{(x-3)^{7}} $$

Step 3: Substitute the rewritten term back into the expression

Replace \((x-3)^{-7}\) in the given expression with \(\frac{1}{(x-3)^{7}}\) to get: $$ x^{3} y^{2}(x-3)^{-7} = x^{3} y^{2} \times \frac{1}{(x-3)^{7}} $$

Step 4: Simplify the expression

The final expression simplified with only positive exponents is: $$ x^{3} y^{2}(x-3)^{-7} = \frac{x^{3}y^{2}}{(x-3)^{7}} $$

Key Concepts

Understanding Negative Exponents

In mathematics, an exponent indicates how many times a number is multiplied by itself. When you encounter a negative exponent, such as \((x-3)^{-7}\), it implies that instead of multiplying, you will divide.
Basically, a negative exponent means "take the reciprocal of the base and then apply the positive exponent." Therefore, \((x-3)^{-7}\) becomes \(\frac{1}{(x-3)^{7}}\).
This conversion is crucial because it allows expressions to be simplified using positive exponents which are often easier to work with in algebraic calculations.

Working with Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators. In the expression \(x^{3} y^{2}(x-3)^{-7}\), you see a product of terms where each contains a variable raised to an exponent.
The key is to treat each part individually:

• \(x^3\) suggests \(x\) multiplied by itself three times.
• \(y^2\) indicates \(y\) squared, or \(y\) multiplied by itself twice.
• And finally, \((x-3)^{-7}\) using a negative exponent, which gets worked on separately as mentioned in the previous section.

Understanding each component helps in rewriting the expression correctly using only positive exponents.

Reciprocal and Its Role in Exponents

A reciprocal of a number is essentially "1 over that number". When converting negative exponents to positive exponents, the reciprocal plays a significant role.
For example, \((x-3)^{-7}\) transforms into \(\frac{1}{(x-3)^{7}}\) through this concept of reciprocation.

The reciprocal is fundamental in exponents because it's the quick step to manage negative exponents, thus allowing for the simplification of algebraic structures into forms that are easier to compute and understand.

The Simplification Process

Simplification is the process of reducing an expression to its most concise form. In the given exercise, simplification involves ensuring all exponents are positive. After rewriting \((x-3)^{-7}\) to \(\frac{1}{(x-3)^{7}}\), you replace it back into the algebraic expression, leading to:

\[ x^{3} y^{2} \times \frac{1}{(x-3)^{7}} \].
You then multiply across to finally get \(\frac{x^{3}y^{2}}{(x-3)^{7}}\), which is the clean, simplified form.

The simplification process makes expressions easier to handle, especially when dealing with complex algebraic problems involving multiple steps or equations.