Problem 57 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(y^{3} z\right)^{1 / 6}}{y^{-1 / 2} z^{1 / 3}} $$

Step-by-Step Solution

Answer

\(\frac{y}{z^{1/6}}\)

Step 1: Apply the Power of a Power Rule

Using the rule \((a^m)^n = a^{m imes n}\), apply this to the numerator \((y^3 z)^{1/6}\). This results in simplification of each term separately: \(y^{3/6} z^{1/6}\), which simplifies to \(y^{1/2} z^{1/6}\).

Step 2: Rewrite the Denominator with Positive Exponents

Since we need to write using positive exponents, rewrite \(y^{-1/2}\) in the denominator as \(\frac{1}{y^{1/2}}\). The denominator becomes \(\frac{1}{y^{1/2} z^{-1/3}}\).

Step 3: Use the Property of Negative Exponents

Using \(a^{-n} = \frac{1}{a^n}\), rewrite \(z^{-1/3}\) as \(\frac{1}{z^{1/3}}\). Thus, the denominator becomes \(\frac{1}{y^{1/2} z^{1/3}}\).

Step 4: Simplify the Whole Expression

Combine the rewritten numerator and denominator. Multiply \(y^{1/2} z^{1/6}\) with the reciprocal of the denominator \((y^{1/2} z^{1/3})^{-1}\), which yields \(y^{1/2} z^{1/6} \times y^{1/2} z^{-1/3}\).

Step 5: Combine Exponents of Like Bases

Simplify by combining exponents for the like bases: for \(y\), add exponents: \(y^{1/2 + 1/2} = y^1\) and for \(z\), add exponents: \(z^{1/6 - 1/3} = z^{1/6 - 2/6} = z^{-1/6}\). This simplifies to \(yz^{-1/6}\).

Step 6: Write with Positive Exponents

Express \(z^{-1/6}\) with a positive exponent using the rule \(a^{-n} = \frac{1}{a^n}\). The expression becomes \(\frac{y}{z^{1/6}}\).

Key Concepts

Power of a Power Rule

The "power of a power" rule is an essential property of exponents that simplifies complex expressions. It states that when a base with an exponent is raised to another exponent, you multiply the exponents. This can be written as \[(a^m)^n = a^{m \times n} .\]For example, if you have \((y^3z)^{1/6},\)you apply this rule by multiplying each exponent within the parentheses by \(1/6\). So, \[(y^3)^{1/6} = y^{3/6} = y^{1/2} ,\] and \[z^{1/6}\].This simplification makes it easier to manipulate and compare different parts of the expression.

Negative Exponents

Negative exponents can be tricky but they have a straightforward meaning. Essentially, if you have an exponent that's negative, it's like saying take the reciprocal of that base with a positive exponent. This can be expressed as \[a^{-n} = \frac{1}{a^n} .\]For instance, in the expression \(y^{-1/2}\), rewriting it using positive exponents involves turning it into \(\frac{1}{y^{1/2}} .\)This rule allows you to transform complex expressions into simpler ones by eliminating negative exponents, which is often required for further simplification.

Simplifying Expressions

Simplifying expressions is all about making them easier to understand and solve. It involves combining like terms, using properties of exponents, and transforming the expression into a more straightforward form. Consider an expression like \[ \frac{(y^{1/2}z^{1/6})}{y^{1/2}z^{1/3}} .\]Here, you can use properties of exponents to combine terms with the same base. By multiplying through these results, you get \[y^{1/2+1/2} = y^1 \] and \[z^{1/6-1/3} = z^{-1/6} .\]This form is simpler and allows for easier computation or conversion into the final required form.

Positive Exponents

Expressing results with positive exponents is often a requirement in mathematics. Positive exponents represent standard multiplicative factors which are straightforward to interpret. If you end up with a negative exponent in your simplified expression, as in \[z^{-1/6}, \]you need to rewrite it with positive exponents using the reciprocal rule. This resulting in \[\frac{1}{z^{1/6}} ,\]makes it simpler and more conventional. This transformation can be crucial in contexts where only positive exponents are acceptable, ensuring clarity and consistency in mathematical results.