Problem 70 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\left(11 r^{10} s^{-3}\right)^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{121r^{20}}{s^6} \).

Step 1: Apply the Power of a Power Rule

First, apply the power of a power rule, which states that \( (a^m)^n = a^{m \cdot n} \), to each part of the expression \( (11r^{10}s^{-3})^2 \). This means you will multiply the exponents inside the parentheses by 2.

Step 2: Simplify the Coefficient

The coefficient 11 is being squared, so compute \( 11^2 = 121 \). This is the new coefficient outside of the parentheses.

Step 3: Simplify the Exponent of r

Apply the rule to \( r^{10} \): \( (r^{10})^2 = r^{10 \cdot 2} = r^{20} \).

Step 4: Simplify the Exponent of s

Similarly, apply the rule to \( s^{-3} \): \( (s^{-3})^2 = s^{-3 \cdot 2} = s^{-6} \).

Step 5: Remove Negative Exponents

To remove the negative exponent for \( s \), rewrite \( s^{-6} \) as \ \frac{1}{s^6} \. This places s in the denominator.

Step 6: Write the Final Simplified Expression

Combine all parts to form the final simplified expression without negative exponents: \[ \frac{121r^{20}}{s^6} \]

Key Concepts

Power of a Power Rule

Let's dive into the "Power of a Power Rule". This fundamental rule in exponentiation helps us understand how to simplify expressions where an exponent is raised to another power. The rule is straightforward: \((a^m)^n = a^{m \cdot n}\). This rule tells us that when you have an exponent raised to another exponent, you simply multiply the two exponents together.

• Example: If you have \((x^3)^2\), you multiply the exponents: \(x^{3 \cdot 2} = x^6\).
• This rule applies to each part of an expression within parentheses. In our exercise, \((11r^{10}s^{-3})^2\), each part inside the parentheses receives this treatment.

This powerful rule simplifies complex expressions and is a vital tool for algebraic manipulations. By applying it, our expression becomes more manageable and easier to work with.

Negative Exponents

Negative exponents can be tricky, but they're quite simple once you grasp the basic concept. A negative exponent means the reciprocal of the base raised to the positive exponent. In mathematical terms, \(a^{-n} = \frac{1}{a^n}\). This tells us to flip the base into the denominator of a fraction.

• Example: \(2^{-3}\) means \(\frac{1}{2^3} = \frac{1}{8}\).
• Applying this to our expression, \(s^{-6}\), we turn it into \(\frac{1}{s^6}\).

Using negative exponents allows us to easily navigate expressions and convert them into a simpler form that avoids direct use of negative exponents, making calculations and further simplification straightforward.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra that involves making calculations more straightforward and manageable. It usually involves combining like terms, reducing fractions, and applying the rules of exponents. For our exercise \, the goal was to simplify \, \((11r^{10}s^{-3})^2\) by performing a series of steps:

• Apply the power of a power rule to each term, multiplying their exponents by 2.
• Simplify the coefficient by squaring it, giving us 121.
• Convert any negative exponents by placing them in the denominator to avoid negatives.

The final simplified expression, \(\frac{121r^{20}}{s^6}\), is free of negative exponents and combines all steps seamlessly. This method efficiently brings complex algebraic expressions to their simplest form, enabling easier interpretation and further mathematical operations.