Problem 595 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{1}{z^2}\).

Step 1: Identify the powers of the variable z

The given expression is \(\frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}}\). Observe that both the numerator and the denominator are powers of the same variable, z.

Step 2: Apply the quotient rule for exponents

Use the quotient rule for exponents which says \(\frac{x^a}{x^b} = x^{a-b}\). Here, \(a = \frac{2}{3}\) and \(b = \frac{8}{3}\). Substitute these values to get \(z^{\frac{2}{3} - \frac{8}{3}}\).

Step 3: Simplify the exponent

Subtract the exponents: \(\frac{2}{3} - \frac{8}{3} = \frac{2 - 8}{3} = \frac{-6}{3} = -2\). Therefore, the expression simplifies to \(z^{-2}\).

Step 4: Convert the negative exponent

Recall that \(x^{-a} = \frac{1}{x^a}\). So, \(z^{-2} = \frac{1}{z^2}\).

Key Concepts

quotient rule for exponents

In algebra, the quotient rule for exponents helps to simplify expressions where a variable with one exponent is divided by the same variable raised to another exponent. The rule states: \[\frac{x^a}{x^b} = x^{a-b} \].

To use the quotient rule, follow these simple steps:

• Identify the base variable which is the same in both the numerator and the denominator.
• Subtract the exponent in the denominator from the exponent in the numerator.
• Rewrite the base with the new exponent.

For example, \[\frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}} \] becomes \[z^{\frac{2}{3} - \frac{8}{3}} = z^{\frac{-6}{3}} = z^{-2} \].

This rule is useful because it helps simplify complex exponential expressions quickly.

negative exponents

Negative exponents might seem tricky at first, but they follow simple rules. A negative exponent means you take the reciprocal of the base and then apply the positive exponent.

Mathematically, this is shown as: \[ x^{-a} = \frac{1}{x^a} \].

For instance, consider the expression \[ z^{-2} \]. To simplify this using the rule for negative exponents:

• Take the reciprocal of the base: \[ z \] becomes \[ \frac{1}{z} \].
• Apply the positive exponent: \[ z^{-2} = \frac{1}{z^2} \].

Now the simplification is clear: \[ z^{-2} \] converts to \[ \frac{1}{z^2} \].

Understanding negative exponents is crucial because they appear in many mathematical contexts and can greatly simplify seemingly complex expressions.

simplification of rational expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying these expressions involves reducing them to their simplest form.

Follow these steps for a smooth simplification:

• Factor both the numerator and the denominator, if possible. Look for common factors that can be canceled out.
• Use the rules of exponents to combine or reduce terms. For example, the quotient rule for exponents can be applied to terms with the same base.
• Simplify any negative exponents by converting them to positive exponents through reciprocals.

In our exercise, we simplified the rational expression \[ \frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}} \] by:

• Applying the quotient rule to get \[ z^{\frac{2}{3} - \frac{8}{3}} = z^{-2} \].
• Converting the negative exponent to get \[ \frac{1}{z^2} \].

This demonstrates how combining these concepts can make simplification straightforward and manageable. Rational expressions become less daunting when you apply these rules systematically.