Problem 41 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ (r s)^{3}(2 s)^{-2}(4 r)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(64r^7s\).

Step 1: Expand Each Term Separately

First, we need to distribute the exponents inside each parenthesis.- \((r s)^{3} = r^{3} s^{3}\)- \((2 s)^{-2} = 2^{-2} s^{-2}\)- \((4 r)^{4} = 4^{4} r^{4}\)

Step 2: Calculate Numerical Values

Simplify the numerical exponents:- \(2^{-2} = \frac{1}{4}\) (since any number raised to a negative exponent is the reciprocal of that number raised to the positive exponent)- \(4^{4} = 256\) (because \(4 \times 4 \times 4 \times 4 = 256\))

Step 3: Combine and Simplify

Now combine all expressions from Step 1 and Step 2:\(r^{3} s^{3} \cdot \frac{1}{4} s^{-2} \cdot 256 r^{4}\) which simplifies to\(\frac{256}{4} \cdot r^{3+4} \cdot s^{3-2}\)

Step 4: Perform Final Simplifications

Carry out the division and simplification:- \(\frac{256}{4} = 64\)- Combine exponents for \(r\): \(r^{3+4} = r^{7}\)- Combine exponents for \(s\):\(s^{3-2} = s^{1} = s\)

Step 5: Write the Final Expression

Combine these simplified elements to get the final expression:\(64r^{7}s\)

Key Concepts

Understanding Negative Exponents

Negative exponents can be confusing, but they're simpler when broken down. When you see a negative exponent, think of it as a way to express division instead of multiplication. For example, if you have a base raised to a negative exponent, such as \(x^{-n}\), it’s equivalent to \(\frac{1}{x^{n}}\). This means you’re taking the reciprocal of the base raised to a positive exponent. Let’s apply this concept to an example:

• \((2s)^{-2}\) becomes \(\frac{1}{(2s)^{2}}\), which simplifies to \(\frac{1}{4s^{2}}\) since \(2^2 = 4\).

By rewriting negative exponents as reciprocals, you make calculations easier and eliminate any negative signs. It’s a useful technique for simplifying expressions.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more manageable form without changing their value. This often means combining like terms or factoring. In algebraic expressions, simplify by eliminating unnecessary components and combining similar elements. Consider the expression after we expand it:

• \(r^{3} s^{3} \cdot \frac{1}{4} s^{-2} \cdot 256 r^{4}\)

To simplify, calculate any numerical values and combine similar terms:

• Find numerical value: \(\frac{256}{4} = 64\)
• Combine the \(r\) terms: \(r^{3} \cdot r^{4} = r^{3+4} = r^{7}\)
• Combine the \(s\) terms: \(s^{3} \cdot s^{-2} = s^{3-2} = s^{1} = s\)

The simplified expression is much cleaner and easier to work with: \(64r^{7}s\). This expression has no negative exponents and no repetitive terms, which is the goal of simplification.

Multiplication of Exponents

The multiplication of exponents is straightforward once you understand the basics. When you multiply expressions with the same base, you add the exponents. For example, with \(a^{m} \times a^{n} = a^{m+n}\). This rule simplifies calculations.Let’s see how this applies to our expression:

• When multiplying \(r^{3}\) and \(r^{4}\), add the exponents: \(r^{3} \cdot r^{4} = r^{3+4} = r^{7}\).
• Similarly, for \(s^{3}\) and \(s^{-2}\), add exponents: \(s^{3} \cdot s^{-2} = s^{3-2} = s^{1} = s\).

By using this rule, complex expressions become easier to handle. Remember, this addition of exponents only applies when bases are the same. The final expression is clear and shows how powerful these simple rules are in handling algebraic expressions efficiently.