Problem 100 - Algebra A Combined Function

Question

Explain how writing \(2 x^{-5}\) with positive exponents is similar to writing \(2 x^{-3 / 4}\) with positive exponents.

Step-by-Step Solution

Answer

Both expressions use the rule of negative exponents: transform \(x^{-n}\) into \(\frac{1}{x^n}\).

Step 1: Understanding Negative Exponents

Negative exponents indicate that the base should be taken as the reciprocal of the base raised to the corresponding positive exponent. For instance, for any non-zero number \(a\), \(a^{-n} = 1/a^n\).

Step 2: Rewriting \(2x^{-5}\) with Positive Exponents

Apply the rule of negative exponents: \(2x^{-5} = 2 \cdot x^{-5} = \frac{2}{x^5}\). The \(x^{-5}\) term is transformed into \(\frac{1}{x^5}\), and we multiply by 2.

Step 3: Rewriting \(2x^{-3/4}\) with Positive Exponents

Similarly, apply the rule to \(2x^{-3/4}\): \(2x^{-3/4} = 2 \cdot x^{-3/4} = \frac{2}{x^{3/4}}\). Here, \(x^{-3/4}\) becomes \(\frac{1}{x^{3/4}}\) while the coefficient remains as 2.

Step 4: Comparing The Two Expressions

Each expression uses the same principle of transforming negative exponents into positive by taking the reciprocal: \(x^{-5}\) becomes \(\frac{1}{x^5}\) and \(x^{-3/4}\) becomes \(\frac{1}{x^{3/4}}\). This makes both expressions similar in the manner they are rewritten.

Key Concepts

Exponent Rules

Exponent rules are fundamental in understanding how to manipulate expressions involving powers. One key rule is that negative exponents indicate the reciprocal of the base. For example, for any number \(a\), if you have \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). This means that instead of multiplying \(a\) by itself, you divide 1 by \(a^n\).
Another important rule is when multiplying powers with the same base, you add their exponents. For instance, \(a^m \times a^n = a^{m+n}\). When dividing, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). These rules are essential for simplifying and rewriting expressions.

• Negative exponents: Reciprocal of the positive exponent version
• Product of powers: Add exponents
• Quotient of powers: Subtract exponents

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operational symbols. They vary in complexity from simple monomials like \(3x\), to polynomials such as \(3x^2 - 5x + 2\). When working with negative exponents, expressions become more dynamic yet require careful handling to maintain equality.
Take for instance the expression \(2x^{-5}\). Here, \(2\) is the coefficient and \(x^{-5}\) involves a negative exponent. By applying the negative exponent rule, we convert \(x^{-5}\) to \(\frac{1}{x^5}\), resulting in \(\frac{2}{x^5}\). This transformation maintains the expression's value while rewriting it in a form that uses positive exponents.

• Coefficient: The numerical factor in front of variables
• Variables: Symbols, often \(x, y\), representing unknown quantities
• Simplification: Converting expressions to a more manageable form using math rules

Reciprocals

Reciprocals are a key concept when dealing with negative exponents. In simple terms, the reciprocal of a number \(a\) is \(\frac{1}{a}\). This idea directly applies to understanding negative exponents. If you have \(x^{-n}\), you're essentially finding the reciprocal of \(x^n\), expressed as \(\frac{1}{x^n}\).
This transformation is vital for expressions like \(2x^{-5}\), which becomes \(\frac{2}{x^5}\) by taking the reciprocal of \(x^5\). This operation keeps the expression's integrity while removing the negative exponent. By applying this reciprocal concept, you ensure the final work aligns with the standard format of expressions using positive exponents only.

• Basic reciprocal: \(\frac{1}{a}\) for a number \(a\)
• Application with exponents: \(x^{-n} = \frac{1}{x^n}\)

Fractional Exponents

Fractional exponents introduce another layer of complexity, representing both radical and exponential operations. A fractional exponent such as \(x^{\frac{m}{n}}\) indicates the \(n\)-th root of \(x^m\). Splitting a fractional exponent into its component indicates roots and powers together.
For example, \(x^{\frac{1}{2}}\) translates to the square root of \(x\), and \(x^{\frac{3}{4}}\) means the fourth root of \(x^3\). When dealing with negative fractional exponents, you first find the reciprocal to handle the negative power, as in \(x^{-\frac{3}{4}} = \frac{1}{x^{\frac{3}{4}}}\). This understanding is crucial for correctly simplifying expressions.

• Interpreting fractional exponents: Combines roots and exponents
• Conversion from fractional to root form: \(x^{\frac{m}{n}}\) as the \(n\)-th root of \(x^m\)
• Handling negative fractional exponents: Reciprocal of the fractional exponent form