Problem 112 - College Algebra

Question

Simplify each exponential expression.Assume that variables represent nonzero real numbers. $$\left(\frac{x^{4} y^{5} z^{6}}{x^{-4} y^{-5} z^{-6}}\right)^{-4}$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \( 1/(x^{32} y^{40} z^{48}) \)

Step 1: Application of Power Rule

The power of a power rule states that \( (a^m)^n = a^{mn} \). Therefore, the expression becomes \( x^{4*(-4)} y^{5*(-4)} z^{6*(-4)} / x^{-4*(-4)} y^{-5*(-4)} z^{-6*(-4)} \)

Step 2: Solving the Exponents

By multiplying the powers, we get \( x^{-16} y^{-20} z^{-24} / x^{16} y^{20} z^{24} \)

Step 3: Application of Quotient Rule

The quotient of powers rule states that \( a^m / a^n = a^{m-n} \). Therefore, the expression becomes \( x^{-16-16} y^{-20-20} z^{-24-24} \)

Step 4: Solving the Last Exponents

By subtracting the powers, we obtain \( x^{-32} y^{-40} z^{-48} \)

Step 5: Application of Negative Exponent rule

\( a^{-m} = 1 / a^m \), applying this rule we have \( 1/x^{32} 1/y^{40} 1/z^{48} \)

Step 6: Final Form

The final simplified form of the expression is thus \( 1/(x^{32} y^{40} z^{48}) \)

Key Concepts

Power Rule

The power rule is a fundamental concept in mathematics, particularly in dealing with exponential expressions. It's all about raising an exponent to another exponent. According to the power rule, when you have a power raised to another power, you simply multiply the exponents. In mathematical terms, this is expressed as \((a^m)^n = a^{m \cdot n}\). This rule helps simplify expressions that would otherwise be quite complex.

For example, if you have \((x^4)^{-4}\), you apply the power rule by multiplying the exponent 4 with -4, giving you \(x^{-16}\). This simplification makes it easier to handle and understand exponential expressions. It is especially useful when dealing with equations that involve variables and exponents to different powers.

Quotient Rule

The quotient rule is another essential principle used in simplifying exponential expressions. It helps when you are dividing numbers with the same base but different exponents. The rule states that you subtract the exponents: \(a^m / a^n = a^{m-n}\).

Consider the expression \(x^{-16}/x^{16}\). By using the quotient rule, you subtract the exponents: \(-16 - 16 = -32\). This results in \(x^{-32}\). Simplifying these expressions is crucial, especially when you need to derive a cleaner form or solve equations involving numerous variable parts.

This rule can also further be applied on multiple variables simultaneously, like in \(y^{-20}/y^{20}\), resulting in \(y^{-40}\) using the same concept.

Negative Exponent Rule

The negative exponent rule is very handy for simplifying expressions with negative exponents. It states that an expression with a negative exponent can be rewritten as a fraction with a positive exponent in the denominator. In other words, \(a^{-m} = 1/a^m\).

For instance, when you encounter an expression like \(x^{-32}\), applying the negative exponent rule transforms it into \(1/x^{32}\). This transformation simplifies calculations and simplifies the expression for further manipulation.

Applying this rule across various variables leads to expressions like \(1/y^{40}\) and \(1/z^{48}\) from \(y^{-40}\) and \(z^{-48}\) respectively. This rule is the final step to rewriting complex expressions into a simplified and most readable form.