Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\frac{7 x^{2}}{x^{3}}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{7 x^{2}}{x^{3}}\) is \(7/x\).

Step 1: Identifying the Terms

Initially, the expression to be solved is \(\frac{7 x^{2}}{x^{3}}\). Here, the term in the numerator is \(7x^{2}\) and the term in the denominator is \(x^{3}\).

Step 2: Applying the Laws of Exponents

The rule for division says that when you divide like bases, you subtract the exponents. So, if we apply the law of exponents for division \(a^{m}/a^{n} = a^{m-n}\), where m is the exponent of term in the numerator and n is exponent of term in the denominator, to our problem, we have \(x^{2}/x^{3} = x^{2-3}\).

Step 3: Subtracting the Exponents

Subtracting 3 from 2 gives us -1. So, the simplified form of \(x^{2}/x^{3} = x^{-1}\). Hence, our expression simplifies to \((7)*x^{-1}\), where * is the multiplication symbol.

Step 4: Final Simplification

The expression \((7)*x^{-1}\) is the same as \(7/x = 7x^{-1}\). We rewrite the factor with the negative exponent as a denominator with a positive exponent. So, the simplification gives us \(7/x\).

Key Concepts

Laws of Exponents

Exponents are a fundamental part of algebra. They allow us to express repeated multiplication in a concise way. The laws of exponents help us simplify expressions with exponents easily:

• Product of powers: Multiply the coefficients, and add the exponents if the bases are the same. \(a^m \cdot a^n = a^{m+n}\).
• Quotient of powers: Divide the coefficients and subtract the exponents. \(a^m / a^n = a^{m-n}\).
• Power of a power: Multiply the exponents \((a^m)^n = a^{mn}\).

For the given problem \( \frac{7 x^2}{x^3} \), we see that both the numerator and the denominator have the base \(x\). Using the quotient law, we set \(m = 2\) and \(n = 3\), which simplifies to \(x^{-1}\). This rule is essential when dealing with expressions that involve division of like bases.

Negative Exponents

Negative exponents might confuse at first, but they're straightforward once you get the hang of them. A negative exponent tells us that we should find the reciprocal of the base and turn the exponent into positive. For instance, \(a^{-n} = \frac{1}{a^n}\). This means:

• Flip the base: If you have \(x^{-1}\), it becomes \(\frac{1}{x}\).
• From numerator to denominator: Negative exponents can move terms from the numerator to the denominator. For example, in our expression, \(7x^{-1}\) becomes \(\frac{7}{x}\).

Understanding negative exponents deepens your comprehension of how terms rearrange themselves in expressions and equations, especially in simplifying rational expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying rational expressions means making them easier to work with by reducing them to their simplest form. Here's where our prior knowledge of exponents comes handy:

• Simplification: Just like numerical fractions, divide common factors from the numerator and denominator.
• Exponents and variables: Laws of exponents simplify variables across the fraction bar (e.g., \(\frac{x^m}{x^n}\) becomes \(x^{m-n}\)).
• Negative exponents: Shift terms with negative exponents as we simplify. In our exercise, \(x^{-1}\) leads us to represent the expression as \(\frac{7}{x}\).

Simplifying rational expressions trains us to .combine all related rules and assemble a cleaner and more straightforward form. Mastering these concepts aid significantly in solving complex algebraic equations and understanding higher-degree mathematics.