Problem 71 - Algebra 1: Concepts and Skills

Question

Rewrite the expression \(\frac{3 x^{-2}}{y^{3} z^{-1}}\) with positive exponents. F. \(\frac{z}{3 x^{2} y^{3}}\) G. \(\frac{3 z}{x^{2} y^{3}}\) H. \(\frac{3 y^{3} z}{x^{2}}\) J. \(3 x^{2} y^{3} z\)

Step-by-Step Solution

Answer

The rewritten expression is \(\frac{3}{x^{2} y^{3} z}\), so the correct choice is G. \(\frac{3z}{x^{2} y^{3}}\)

Step 1: Identify the terms with negative exponents

In the expression \(\frac{3 x^{-2}}{y^{3} z^{-1}}\), the terms with negative exponents are \(x^{-2}\) and \(z^{-1}\).

Step 2: Rewrite the terms with negative exponents using positive exponents

Using the rule that \(a^{-n} = \frac{1}{a^{n}}\), the terms can be rewritten as \(\frac{1}{x^{2}}\) and \(\frac{1}{z}\).

Step 3: Substitute the rewritten terms back into the expression and simplify

Substituting the rewritten terms, the expression becomes \(\frac{3}{x^{2} y^{3} z}\). Check all the options to see which option matches this arrangement.

Key Concepts

Understanding Negative Exponents

Negative exponents in algebra are a way of expressing division operations in an alternative form. When you encounter an exponent that is negative, such as in the term \( x^{-2} \), it signifies that you need to take the reciprocal of the base raised to the absolute value of the exponent. In other words, \( x^{-2} \) is the same as \( \frac{1}{x^2} \). This transformation is a crucial aspect of simplifying algebraic expressions, as it helps in converting all exponents to positive ones, which makes it easier to manipulate or combine like terms.

Let's investigate the example \( \frac{3 x^{-2}}{y^{3} z^{-1}} \), which includes negative exponents. To simplify the expression, we apply the reciprocal rule:

• \( x^{-2} \) becomes \( \frac{1}{x^{2}} \)
• \( z^{-1} \) becomes \( \frac{1}{z} \)

Remember, the goal is always to express the algebraic expression in its simplest form with only positive exponents.

Exponent Rules to Remember

When simplifying algebraic expressions, knowing the rules of exponents is critical. Here are some of the basic rules that are essential:

• The Product Rule: \( a^m \cdot a^n = a^{m+n} \)
• The Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• The Power Rule: \( (a^m)^n = a^{m \cdot n} \)
• The Zero Exponent Rule: \( a^0 = 1 \), where \( a \) is not zero.
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)

Applying these rules simplifies complex expressions and allows you to restructure terms accordingly. For example, the Negative Exponent Rule is used to transform negative exponents into positive ones, which we've used in our current problem to simplify \( x^{-2} \) and \( z^{-1} \). With these rules, you can adeptly manage various algebraic operations and solve problems with ease.

Mastering Algebraic Manipulation

Algebraic manipulation involves rearranging algebraic expressions to a simpler or more useful form. This often includes factoring, expanding, and simplifying terms. To achieve this, you need to combine the exponent rules with basic algebra skills such as distributing, combining like terms, and canceling common factors.

In our example, once the negative exponents are converted, we continue to simplify the expression. Since \( \frac{3}{x^{2} y^{3} z} \) is already quite simplified, no further algebraic manipulation is necessary. It’s important to always look for like terms to combine and for any opportunities to simplify fractions.

When you're faced with a complex expression, remember to:

• Identify like terms and any available common factors.
• Use the appropriate exponent rules to combine or simplify terms with exponents.
• Apply basic arithmetic operations to consolidate the expression.

Algebraic manipulation is a powerful tool that helps you not just in simplifying expressions, but also in solving equations and understanding the structure of algebraic functions.