Problem 84 - Elementary Algebra

Question

Simplify: \(\left(\frac{z^{7}}{z^{4}}\right)^{2}\) a. Use the quotient rule first. Then use the power rule. b. Use the many bases rule for a quotient first. Next, use the power rule. Finally, use the quotient rule. c. Compare the simplified expressions from part a and part b.

Step-by-Step Solution

Answer

The simplified expression is \(z^6\) using both methods.

Step 1 (Part a) - Apply the Quotient Rule

Use the Quotient Rule for exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\), to the expression \(\frac{z^7}{z^4}\). This gives \(z^{7-4} = z^3\).

Step 2 (Part a) - Apply the Power Rule

Now apply the Power Rule, which states that \((a^m)^n = a^{mn}\), to \(z^3\), raised to the power of 2. This results in \(z^{3 \times 2} = z^6\).

Step 1 (Part b) - Many Bases Rule for a Quotient

First, apply the rule of distributing an exponent over a quotient, which states that \((\frac{a}{b})^m = \frac{a^m}{b^m}\). This changes the expression to \( \frac{(z^7)^2}{(z^4)^2} \).

Step 2 (Part b) - Apply the Power Rule

Next, apply the Power Rule, which states that \((a^m)^n = a^{mn} \), to both the numerator and the denominator. This changes the expression to \(\frac{z^{7 \times 2}}{z^{4 \times 2}} = \frac{z^{14}}{z^8} \).

Step 3 (Part b) - Apply the Quotient Rule

Finally, apply the Quotient Rule, which states that \(\frac{a^m}{a^n} = a^{m-n} \), to \(\frac{z^{14}}{z^8}\). This results in \(z^{14-8} = z^6 \).

Step 1 (Part c) - Compare the Simplified Expressions

Both methods (Part a and Part b) lead to the same simplified expression: \(z^6\).

Key Concepts

Quotient Rule

The quotient rule is a fundamental concept in algebra, especially for simplifying expressions involving exponents. It states that for any nonzero number \( a \) and any integers \( m \) and \( n \), the expression \( \frac{a^m}{a^n} \) can be simplified to \( a^{m-n} \).
This means that when you divide two powers with the same base, you can subtract the exponent of the denominator from the exponent of the numerator.
For instance:

• \( \frac{x^5}{x^2} = x^{5-2} = x^3 \)
• \( \frac{y^9}{y^3} = y^{9-3} = y^6 \)

Let's apply this to the exercise:

• Start with the given expression \( \left( \frac{z^7}{z^4} \right)^2 \).
• Using the quotient rule: \( \frac{z^7}{z^4} = z^{7-4} = z^3 \).

So after applying the quotient rule, the expression simplifies to \( z^3 \).

Power Rule

The power rule is another essential tool in algebra, particularly for dealing with exponents. This rule states that when you raise a power to another power, you multiply the exponents: \( (a^m)^n = a^{mn} \).
This means if you have a base with an exponent, and then raise that to another exponent, you combine the exponents through multiplication.
Examples include:

• \( (b^3)^2 = b^{3 \times 2} = b^6 \)
• \( (c^4)^3 = c^{4 \times 3} = c^{12} \)

Returning to the exercise and continuing from the quotient rule result:

• We have \( z^3 \), which is then raised to the power of 2.
• Using the power rule: \( (z^3)^2 = z^{3 \times 2} = z^6 \).

So, after applying the power rule to \( z^3 \), the simplified expression is \( z^6 \).

Exponent Rules

Understanding the basic exponent rules is crucial for tackling complex algebraic expressions. These rules include the quotient rule, the power rule, and an additional rule for distributing exponents over a quotient.
This last rule states: \( \left( \frac{a}{b} \right)^m = \frac{a^m}{b^m} \).
This means that when you have a quotient (division) raised to an exponent, the exponent applies individually to both the numerator and the denominator.
Example:

• \( \left( \frac{3}{2} \right)^4 = \frac{3^4}{2^4} = \frac{81}{16} \)

Applying this to the exercise:

• Start with \( \left( \frac{z^7}{z^4} \right)^2 \).
• Distribute the exponent: \( \frac{(z^7)^2}{(z^4)^2} \).

Now apply the power rule to both numerator and denominator:

• \( (z^7)^2 = z^{7 \times 2} = z^{14} \)
• \( (z^4)^2 = z^{4 \times 2} = z^8 \)

This simplifies the expression to \( \frac{z^{14}}{z^8} \).
Finally, using the quotient rule again:

• \( \frac{z^{14}}{z^8} = z^{14-8} = z^6 \).

Thus, all steps lead to the same final result: \( z^6 \).