Problem 30 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(-2 r^{2} s\right)^{5}\left(3 r^{-1} s^{3}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(-288r^8s^{11}\).

Step 1: Distribute Exponents Across Terms

First, apply the exponents to each term inside the parentheses: \[(-2r^2s)^5 = (-2)^5(r^2)^5(s)^5\]\[(3r^{-1}s^3)^2 = (3)^2(r^{-1})^2(s^3)^2\]This distributes the exponents to each factor inside the parentheses.

Step 2: Evaluate Powers of Numbers

Calculate the powers of the numerical coefficients:\[(-2)^5 = -32\] \[(3)^2 = 9\]This gives us the numerical part of each expression.

Step 3: Apply Power Rules to Variables

Using the power of a power rule \((x^a)^b = x^{ab}\), simplify the variable terms:For \((r^2)^5\): \[r^{2 imes 5} = r^{10}\]For \((s)^5\): \[s^{5}\]For \((r^{-1})^2\): \[r^{-2}\]For \((s^3)^2\): \[s^{6}\]These calculations simplify the variable terms to a single set of powers.

Step 4: Combine Like Terms

Now multiply the results of the two expressions together:\[(-32)r^{10}s^{5} \cdot 9r^{-2}s^{6}\]Combine the numerical coefficients: \[-32 \times 9 = -288\]Add the exponents of like bases:For \(r\): \[r^{10} \times r^{-2} = r^{10 + (-2)} = r^{8}\]For \(s\): \[s^{5} \times s^{6} = s^{5 + 6} = s^{11}\]Hence, the expression becomes \(-288r^8s^{11}\).

Key Concepts

Exponentiation

Exponentiation is an essential concept in algebra. It involves raising a number or variable to a certain power, which means multiplying it by itself a specific number of times. When you see something like \[x^m\], it means the base \(x\) is multiplied by itself \(m\) times.
For example, consider the expression \((-2)^5\). Here, \(-2\) is multiplied by itself five times:

• \((-2) \times (-2) \times (-2) \times (-2) \times (-2)\)

The result is \(-32\), as five factors of \(-2\) combine to give this product. Exponentiation helps simplify repeated multiplication.Working with variables, such as in \((r^2)^5\), requires a similar process. Instead of multiplying numbers, you're multiplying a variable power by itself, as shown through the Power Rules. This operation simplifies complex expressions.

Simplification

Simplification is the process of making an algebraic expression easier to work with. When you simplify an expression, you are reducing it to its most basic form without changing its value. The goal is to eliminate any unnecessary complexity.
In our exercise, we've begun with:

• \((-2r^2s)^5(3r^{-1}s^3)^2\)

The task is to simplify this to something straightforward. Simplifying involves tasks such as distributing exponents across inside terms or calculating powers of coefficients—like reducing \((-2)^5\) to \(-32\) and \((3)^2\) to \(9\).Simplification includes breaking down power terms into more manageable components using techniques from different rules such as Power Rules and Combining Like Terms. By following the step-by-step reduction, we arrive at the simplest form like we did with the final expression \(-288r^8s^{11}\).

Power Rules

Power Rules help simplify expressions with exponents. A critical rule is the 'power of a power' rule, which involves taking an exponentiated term raised to another power and multiplying the powers:
\[(x^a)^b = x^{ab}\]In the exercise:

• \((r^2)^5\) becomes \(r^{10}\) because \(2 \times 5 = 10\).
• \((r^{-1})^2\) simplifies to \(r^{-2}\) since \(-1 \times 2 = -2\).
• \((s^3)^2\) results in \(s^{6}\) through the calculation \(3 \times 2 = 6\).

Power Rules allow for streamlined simplification by enabling the combination of like bases through multiplication of their exponents, which reduces the complexity of expressions and prepares them for further reduction.

Combining Like Terms

Combining Like Terms is a fundamental algebraic process. It involves simplifying expressions by adding or subtracting terms that have the same variable base raised to the same power. In algebra, a 'term' refers to a product of constants and variables.Consider the expressions:

• \((-32)r^{10}s^5\)
• \(9r^{-2}s^6\)

By multiplying these, we must combine terms with like bases. For instance, with \(r\), we simplify:

• \(r^{10} \times r^{-2} = r^{8}\)

Here, exponents are added: \(10 + (-2) = 8\).For \(s\), the combination is:

• \(s^{5} \times s^{6} = s^{11}\)

Adding exponents \(5 + 6 = 11\) results in \(s^{11}\).By using this method, complex expressions are blended into a more compact form. In this exercise, the combined expression simplifies to \(-288r^8s^{11}\), illustrating how merging like terms helps solve complex algebraic problems.