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Chapter 1: Problem 30

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

Short Answer

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

Step by step solution

01

Apply Power to Each Term

When a term with multiple variables or constants is raised to a power, apply the power to each factor inside the term. For the expression \((-2 r^{2} s)^{5}\), apply the power of 5 to -2, \(r^{2}\), and \(s\). Similarly, for \((3 r^{-1} s^{3})^{2}\), apply the power of 2 to 3, \(r^{-1}\), and \(s^{3}\).
02

Simplify Each Term Individually

Calculate each term separately: \((-2)^{5} = -32\), \((r^{2})^{5} = r^{10}\), and \((s)^{5} = s^{5}\) for the first expression. For the second expression, \((3)^{2} = 9\), \((r^{-1})^{2} = r^{-2}\), and \((s^{3})^{2} = s^{6}\). This simplifies the expressions to \(-32r^{10}s^{5}\) and \(9r^{-2}s^{6}\).
03

Multiply the Simplified Expressions

Now multiply the two new expressions. Combine like terms: \(-32 imes 9 = -288\), \((r^{10})(r^{-2}) = r^{8}\), and \((s^{5})(s^{6}) = s^{11}\). This gives the final simplified expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions involves reducing a complex expression to its simplest form. This process typically makes it easier to work with the expression in equations or further mathematical problems. Let's break down the simplification:
  • Apply Operations: Look for operations that can be easily calculated such as multiplication or division within the expression.
  • Utilize Mathematical Laws: Use laws such as distributive, associative, or commutative laws to rearrange or reduce expressions.
  • Combine Like Terms: Terms that have the same variables raised to the same powers can be combined by performing the arithmetic operation attached to them, leading to a simpler result.
Whenever you simplify an expression, remember that the goal is to retain the value of the expression, yet make it more concise.
Variable Exponents
Variable exponents occur when variables are raised to a power. Understanding how to manipulate these is crucial for simplifying expressions.
  • Power of a Variable: When you have a situation like \(x^{a}\) and you apply a power to it, \(a\) is multiplied by the new power. For example, \( (x^{2})^{3} = x^{6} \).
  • Multiplying Variables: When multiplying variables with the same base, you add their exponents. For example, \( x^{m} \cdot x^{n} = x^{(m+n)} \).
  • Negative Exponents: These represent reciprocals: \( x^{-n} = \frac{1}{x^{n}} \. \) They flip the base under division.
Variable exponents can be tackled with these simple rules, making the task of simplifying them straightforward once you practice.
Power of a Product Rule
The power of a product rule is a fundamental tool in managing expressions involving products (i.e., terms multiplied together) raised to powers. According to this rule, you can distribute the exponent over the terms within the parenthesis.
  • Applying the Rule: For any product of terms \( (a\cdot b)^{n} \), the rule states \( (a^{n}\cdot b^{n}) \).
  • Step-by-Step Application: This means, each term inside the parenthesis receives the exponent. For example, \( (3ab)^{2} = 3^{2} \cdot a^{2} \cdot b^{2} \).
  • Simplifying Afterwards: Once the rule is applied, work on simplifying each resulting term individually before combining them.
Mastering this rule allows you to break down complex expressions into simpler parts, making them manageable and easy to work with in any algebraic operation.

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Most popular questions from this chapter

Simplify the expression. $$\left(2 x^{2}-3 x+1\right)(4)(3 x+2)^{3}(3)+(3 x+2)^{4}(4 x-3)$$

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{\frac{3 x}{2 y^{3}}}$$

Simplify the expression. $$\frac{\left(x^{2}+2\right)^{3}(2 x)-x^{2}(3)\left(x^{2}+2\right)^{2}(2 x)}{\left[\left(x^{2}+2\right)^{3}\right]^{2}}$$

Rationalize the numerator. $$\frac{\sqrt{2(x+h)+1}-\sqrt{2 x+1}}{h}$$

Simplify the expression. $$\frac{\frac{r}{s}+\frac{s}{r}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$
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