Problem 14 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \frac{6^{-1} x^{3} y^{-5} x^{-3}}{y^{-5}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression in a way that all the exponents are positive: $$\frac{6^{-1} x^{3} y^{-5} x^{-3}}{y^{-5}}$$ Answer: $$\frac{1}{6}$$

Step 1: State the given expression

We have the given expression: $$ \frac{6^{-1} x^{3} y^{-5} x^{-3}}{y^{-5}} $$

Step 2: Use the exponent rules to simplify the expression

First, we will simplify the expression by applying the exponent rules. For any non-zero numbers a and b, and integers m and n, the following rules apply: 1. \(a^m \cdot a^n = a^{m + n}\) 2. \(\frac{a^m}{a^n} = a^{m - n}\) 3. \((a^m)^n = a^{mn}\) Applying rule 1: $$ \frac{6^{-1} x^{3} y^{-5} x^{-3} }{ y^{-5} } = \frac{6^{-1} x^{3-3} y^{-5}}{ y^{-5} } $$ Which simplifies to: $$ \frac{6^{-1} x^{0}y^{-5}}{y^{-5}} $$

Step 3: Apply the exponent rules again

Now, applying rule 2 to simplify the expression: $$ \frac{6^{-1} x^{0}y^{-5}}{y^{-5}} = 6^{-1} x^{0}\cdot y^{-5+5} $$ And this expression simplifies to: $$ 6^{-1} x^{0}y^{0} $$

Step 4: Exponential properties of 0 and 1

We have two special exponential properties that we can use here: 1. \(a^0 = 1\) for any nonzero value of a. 2. \(a^{-n} = \frac{1}{a^n}\) for any nonzero value of a. $$ 6^{-1} x^{0} y^{0} = \frac{1}{6^1} \cdot 1 \cdot 1 $$

Step 5: Write the simplified expression

The given expression simplifies to: $$ \frac{1}{6} $$ Now the expression has been simplified with all the exponents appearing as positive.

Key Concepts

Exponent Rules

Exponent rules are essential for simplifying algebraic expressions that involve powers of variables and constants. By understanding and applying these rules, you can manipulate expressions to make them easier to work with. Here are some of the key rules:

• **Product Rule:** When you multiply two powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m + n}\).
• **Quotient Rule:** When you divide two powers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m - n}\).
• **Power of a Power Rule:** When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\).

These rules make it possible to simplify expressions like the one given: \(\frac{6^{-1} x^3 y^{-5} x^{-3}}{y^{-5}}\). Using these rules, you can easily combine, cancel, or rearrange exponents to make the expression simpler.

Positive Exponents

When working with exponents, it's often more convenient to express them as positive numbers. This is because positive exponents are easier to interpret and work with in equations.

• **Negative Exponent Rule:** \(a^{-n} = \frac{1}{a^n}\). This means that a negative exponent indicates a reciprocal. For instance, \(6^{-1}\) becomes \(\frac{1}{6}\) when expressed with a positive exponent.
• **Zero Exponent Rule:** Regardless of the base (except zero), any number raised to the power of zero equals one: \(a^0 = 1\).

In our example, transitioning from negative exponents to positive exponents helps in simplifying the expression further, providing an answer that's more understandable and easier to work with.

Simplifying Expressions

Simplifying expressions involves using exponent rules and converting all exponents to positive numbers whenever possible. This process reduces the expression to its simplest form, making it easier to handle in equations and analyses.

• First, identify like bases and apply the product and quotient rules to combine or reduce terms.
• Next, switch any negative exponents to positive by moving terms across the fraction line (from numerator to denominator or vice versa).
• Finally, apply any special properties, like \(x^0 = 1\), to reduce terms to their simplest form.

In the given problem, following these steps correctly leads to a completely simplified expression: \(\frac{1}{6}\). Each exponent is now positive, demonstrating the power and utility of these processes in algebra.