Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: 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 \(-288 r^8 s^{11}\).

Step 1: Apply the Power Rule to Each Term in Parentheses

For the expression \((-2 r^{2} s)^{5}\), apply the power rule \(a^m\right)^n = a^{m \cdot n}\) to get:\[(-2)^5 \cdot (r^2)^5 \cdot (s)^5 = -32 r^{10} s^5\].Similarly, for the expression \((3 r^{-1} s^{3})^{2}\), apply the power rule to produce:\[3^2 \cdot (r^{-1})^2 \cdot (s^{3})^2 = 9 r^{-2} s^6\].

Step 2: Multiply the Simplified Expressions

Now multiply the results from Step 1:\[(-32 r^{10} s^5)(9 r^{-2} s^6)\].This can be further broken down as: - Numerically: \(-32 \cdot 9 = -288\).- For \(r\) terms: \(r^{10} \cdot r^{-2} = r^{10 - 2} = r^8\).- For \(s\) terms: \(s^5 \cdot s^6 = s^{5 + 6} = s^{11}\).

Step 3: Write the Final Solution

Combining all the components from Step 2, the final simplified expression is:\[-288 r^8 s^{11}\].

Key Concepts

Power Rule

When you're working with exponents and see something like \(a^m\right)^n = a^{m \cdot n}\), you're using the power rule. This rule is super helpful because it allows you to simplify expressions in a neat way. For example, if you have \((x^4)^3\), applying the power rule means multiplying the exponents: \(4 \, times \, 3\). The result is \(x^{12}\).
You just turn nested powers into a single power.
In the problem we're looking at, the power rule is used on expressions like \(r^2\right)^5\), which simplifies to \(r^{10}\). This keeps the math manageable.

Exponent Laws

Learning exponent laws is crucial because they govern how you multiply, divide, and manipulate powers. Here are some important ones to know:

• Multiplying Powers: \(a^m \cdot a^n = a^{m+n}\). Just add the exponents if the bases are the same.
• Dividing Powers: \(a^m / a^n = a^{m-n}\). Subtract the exponents if the bases are the same.
• Negative Exponents: \(a^{-m} = 1/a^m\). This means the reciprocal of the base.

In our exercise, the exponent laws help us simplify powers of \(-2, r, \, and \, s\). For example, you can see it in \((r^{10} \cdot r^{-2} = r^8)\), by subtracting the exponents.

Polynomial Multiplication

Multiplying polynomials, which expands products like \(ab(ef)\), is about dealing with multiple terms at once. It often uses a distribution method. Here's a simple way to look at it:

• Numerical Multiplication: Combine the coefficients, like \(-32 \, times \, 9 = -288\).
• Same Base Variables: Use exponent laws, multiplying them by adding exponents.

In our simplification case, you multiply two parts: \(-32 r^{10} s^5\) with \(9 r^{-2} s^6\). You deal with numbers first, then the \(r\) variables, and lastly the \(s\) terms.
This keeps your work structured efficiently.

Negative Exponents

Sometimes you encounter exponents with a negative sign, like \((r^{-1})^2\). Here, negative exponents imply reciprocals. For example, \(r^{-2}\) can be rewritten as \((1/r)^2\) or \(1/r^2\).
It makes managing expressions more intuitive.
In the solution, applying this rule helps you simplify the terms step-by-step. \((r^{-1})^2\) becomes \(r^{-2}\). Later, you can combine with other terms using exponent laws. Understanding negative exponents is key when rearranging and simplifying long algebraic statements.