Problem 83 - Elementary Algebra

Question

Simplify: \(\left(\frac{x^{9}}{x^{3}}\right)^{2}\) 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 \( x^{12} \).

Step 1: Apply the Quotient Rule

Use the quotient rule, which states \(\frac{a^m}{a^n} = a^{m-n}\), to simplify \(\frac{x^9}{x^3}\)\. Therefore, \(\frac{x^9}{x^3}=x^{9-3}=x^6\)\.

Step 2: Apply the Power Rule

Use the power rule, which states \( (a^m)^n = a^{mn} \), to simplify \((x^6)^2\). Therefore, \( (x^6)^2 = x^{6*2}=x^{12} \)\.

Step 3: Simplify using Many Bases Rule

For part b, start from \(\frac{x^9}{x^3}\) and simplify it as \( (x^9 \times x^{-3})^2 \). Use \( x^m \times x^n = x^{m+n} \), we get \((x^{9-3})^2 = (x^6)^2 \).

Step 4: Apply the Power Rule Again

Using \( (a^m)^n = a^{mn} \), simplify \((x^6)^2 = x^{12} \)\.

Step 5: Compare Results from Parts a and b

In both methods, the simplified expression is \( x^{12} \).

Key Concepts

Quotient Rule

The quotient rule in algebra helps simplify expressions involving division of the same base with different exponents. It states: \( \frac{a^m}{a^n} = a^{m-n} \).

For instance, in our exercise, we start with \( \frac{x^9}{x^3} \).

Using the quotient rule, we subtract the exponents: \( 9 - 3 \).

Therefore, \( \frac{x^9}{x^3} = x^{9-3} = x^6 \).

This rule is crucial for simplifying algebraic fractions where the bases are identical.

Power Rule

The power rule is another fundamental principle in algebra, used to deal with exponents raised to another power.

It states: \( (a^m)^n = a^{mn} \).

In our example, after using the quotient rule, we have \( x^6 \).

Next, we raise this to the power of 2: \( (x^6)^2 \).

Applying the power rule, we multiply the exponents: \( 6 \times 2 \).

Thus, \( (x^6)^2 = x^{12} \).

This rule simplifies expressions involving multiple layers of exponentiation.

Many Bases Rule

The many bases rule is a helpful approach when dealing with a quotient raised to a power. This method can sometimes make complex expressions more understandable.

In our example, we start by expressing the initial fraction using laws of indices:

\( \frac{x^9}{x^3} = x^9 \times x^{-3} \).

According to the properties of exponents, \( x^m \times x^n = x^{m+n} \).

This gives us \( x^{9-3} = x^6 \).

Now, raising \( x^6 \) to the power of 2 as before, we use the power rule to get: \( (x^6)^2 = x^{12} \).

The many bases rule reinforces the application of both the quotient and power rules in algebra, ensuring the same result through a different strategic process.