Problem 54 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \left(x^{2}+3\right)^{3}\left(x^{2}-1\right)^{-4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left(x^2 + 3\right)^3 \left(x^2 - 1\right)^{-4}$ using only positive exponents. Answer: $\frac{\left(x^2 + 3\right)^3}{\left(x^2 - 1\right)^4}$

Step 1: Identify the term with a negative exponent

The given expression has one term with a negative exponent, which is: $$ \left(x^2 - 1\right)^{-4}. $$

Step 2: Apply the property of negative exponents

Now, we will apply the property of negative exponents to the term with the negative exponent. Using the property: $$ \left(x^2 - 1\right)^{-4} = \frac{1}{\left(x^2 - 1\right)^4}. $$

Step 3: Rewrite the original expression using only positive exponents

Now that we've applied the property of negative exponents to our term, we can rewrite the original expression using only positive exponents: $$ \left(x^2 + 3\right)^3 \left(x^2 - 1\right)^{-4} = \left(x^2 + 3\right)^3 \cdot \frac{1}{\left(x^2 - 1\right)^4}. $$

Step 4: Simplify the expression

Though this part is optional, we can further simplify the expression by writing it as a single fraction: $$ \frac{\left(x^2 + 3\right)^3}{\left(x^2 - 1\right)^4}. $$ This final expression uses only positive exponents and represents the simplified version of the original expression.

Key Concepts

Understanding Negative Exponents

Negative exponents are a way of expressing mathematical expressions that involve division or reciprocals. When you see a term with a negative exponent, like \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). In other words, the negative exponent tells you to take the reciprocal of the base raised to the positive exponent.

• For example, \(x^{-3}\) is the same as \(\frac{1}{x^3}\).
• This is especially useful for simplifying expressions that involve division.

In our exercise, the expression \((x^2 - 1)^{-4}\) was simplified by turning it into a fraction: \(\frac{1}{(x^2 - 1)^4}\). This conversion from a negative to a positive exponent helps in writing expressions in a standardized format.

Simplifying Expressions

Simplifying expressions is an important skill in algebra. It involves reducing expressions to their simplest form without changing their value. By applying the properties of exponents and using rules such as reversing negative exponents and combining like terms, expressions become easier to manage.
Here are some steps you commonly follow when simplifying:

• Look for negative exponents and apply the property \(a^{-n} = \frac{1}{a^n}\).
• Combine like terms—terms that have the same variable and exponent.
• Rewrite the expression in simpler terms, as a single fraction if possible.

In the original problem, the expression was successfully simplified by combining all terms under a single division sign. The expression started as \((x^2 + 3)^3(x^2 - 1)^{-4}\) and ended as \(\frac{(x^2 + 3)^3}{(x^2 - 1)^4}\). This form clearly illustrates the relationships between the terms using only positive exponents.

Properties of Exponents

The properties of exponents provide a set of rules that simplify working with expressions involving powers. Understanding these helps in rewriting and manipulating algebraic expressions conveniently. Here are some critical exponent rules:

• Multiplication of Like Bases: \(a^m \cdot a^n = a^{m+n}\). This adds the exponents together.
• Division of Like Bases: \(a^m / a^n = a^{m-n}\). This subtracts the exponents.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\). This multiplies the exponents together.
• Negative Exponents: \(a^{-n} = \frac{1}{a^n}\).

These properties were essential in solving the original exercise. By recognizing the negative exponent and turning it positive through division, the expression was simplified to its simplest form. The understanding and application of these rules help reduce complex problems to easy-to-manage expressions.