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Chapter 3: Problem 37

Find the unknown factor. $$-36 u^{5}=(?) \cdot\left(-9 u^{4}\right)$$

Short Answer

Expert verified
)\cdot(-9u^4)$$? Answer: The unknown factor is \(4u\).

Step by step solution

01

Write down the given equation

We are given the following equation: $$-36u^5=(?)\cdot(-9u^4)$$
02

Divide both sides by the term (-9u^4)

To find the unknown factor, we will divide both sides of the equation by (-9u^4): $$ \frac{-36u^5}{-9u^4} = \frac{(?)\cdot(-9u^4)}{-9u^4} $$
03

Simplify the equation

When we simplify the above equation, we will find the value of the unknown factor. First, notice that both sides have a common factor of (-9u^4). The left side also has an additional factor that when multiplied by (-9u^4), yields the result on the right side of the equation: $$ \frac{-36u^5}{-9u^4} = \frac{4u}{1} = 4u \Rightarrow (?) $$
04

State the unknown factor

Now that we have found the unknown factor, we can simply state it as the result: The unknown factor is \(4u\).

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Key Concepts

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

Equation Solving
Equation solving is a fundamental part of algebra that allows us to find the unknown variable or factor in a given expression. In the given exercise, we aim to isolate the unknown factor by performing algebraic operations.

Here's a straightforward approach:
  • Identify the equation: In our exercise, it's \(-36u^5 = (?) \cdot (-9u^4)\).
  • Perform the inverse operation: To isolate the unknown factor "(?)," divide both sides by the given multiplier \(-9u^4\).
  • Simplify: Completing this division allows us to simplify and solve for the unknown factor.
By breaking down the problem into these steps, the unknown can be more easily uncovered.
Factors
Factors are numbers or expressions that multiply together to form another number or expression. In algebra, understanding factors helps in simplifying and solving equations.

For the exercise, note the components:
  • \(-36u^5\): This is the product of the multiplication.
  • \(-9u^4\) and \(4u\): When multiplied, these give us \(-36u^5\).
Decomposing large expressions into factors helps us identify relationships between terms and solve for unknowns. Recognizing common factors can simplify computations and uncover hidden patterns in algebraic expressions.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operations. Sometimes, we need to manipulate these expressions to find specific values or unknowns.

In our task, we dealt with expressions like \(-36u^5\) and \(-9u^4\), involving variables raised to exponents. When solving these, remember:
  • Variables: Symbols such as \(u\) that represent unknown values.
  • Exponents: These indicate repeated multiplication of a variable (e.g., \(u^5\) means \(u\) multiplied by itself five times).
  • Operations: Basic mathematical procedures (addition, subtraction, multiplication, division) blend between numbers and variables.
By understanding these components, we tailor our calculations to manage and solve for unknown factors efficiently.

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