Problem 134 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(x^{6} y^{6} z^{-1}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(x^{6} y^{6} z^{-1}\right)^{2}\) so that only positive exponents are present. Answer: \(\frac{x^{12} y^{12}}{z^{2}}\)

Step 1: Apply power of a power rule

To simplify this expression, we first apply the power of a power rule: \((a^m)^n = a^{mn}\). We will apply this rule to each term inside the brackets. $$ \left(x^{6} y^{6} z^{-1}\right)^{2} = x^{6 \cdot 2} y^{6 \cdot 2} z^{-1 \cdot 2} $$

Step 2: Simplify the exponents

Now, we will simply multiply each exponent by 2 to get the simplified exponents: $$ x^{6 \cdot 2} y^{6 \cdot 2} z^{-1 \cdot 2} = x^{12} y^{12} z^{-2} $$

Step 3: Move the term with a negative exponent to the denominator

Since our goal is to have only positive exponents, we will move the term with the negative exponent to the denominator. This will make the exponent positive. $$ x^{12} y^{12} z^{-2} = \frac{x^{12} y^{12}}{z^{2}} $$ The expression with only positive exponents is \(\frac{x^{12} y^{12}}{z^{2}}\).

Key Concepts

Power of a Power Rule

Understanding the "Power of a Power Rule" is crucial in working with exponents. This rule helps us simplify expressions where an exponent is raised to another exponent. For any base \(a\), and exponents \(m\) and \(n\), the rule is expressed as

• \((a^m)^n = a^{m \times n}\)

By applying this rule, we only need to multiply the two exponents together.
For example, in the expression \((x^6)^2\), instead of calculating \(x^6\) and then squaring the result, we multiply the exponents directly to simplify this to \(x^{12}\).
This not only makes calculations more straightforward but is also efficient for simplifying algebraic expressions involving multiple terms with exponents.

Negative Exponents

Negative exponents can be confusing, but they follow a simple logic. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In other words:

• \(a^{-n} = \frac{1}{a^n}\)

When simplifying expressions, we convert negative exponents into positive ones by moving the base to the opposite part of the fraction (numerator to denominator or vice versa).
In our example, \(z^{-2}\) becomes \(\frac{1}{z^{2}}\). This conversion is essential when rewriting expressions with only positive exponents.
Understanding this concept helps in rewriting and manipulating algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and exponents. Simplifying these expressions involves combining like terms and applying rules of exponents like those mentioned earlier. In the given example, the expression \((x^6 y^6 z^{-1})^2\) includes both positive and negative exponents.

• The goal is to express it using only positive exponents.
• This requires applying the power of a power rule and adjusting negative exponents.

Algebra is not just about moving numbers around; it's about understanding the relationships and rules that allow us to represent numbers symbolically. Mastery of algebraic expressions provides a foundation for more complex problem solving and mathematical reasoning.