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Chapter 4: Problem 7

Find each product. $$ 5 y^{4}\left(3 y^{7}\right) $$

Short Answer

Expert verified
15y^{11}

Step by step solution

01

- Simplify the Expression

First, rewrite the given expression to understand it better: \(5 y^{4} \times 3 y^{7}\)
02

- Multiply the Constants

Multiply the numerical coefficients (constants) 5 and 3: \(5 \times 3 = 15\)
03

- Apply the Law of Exponents

Use the law of exponents which states that when you multiply two terms with the same base, you add the exponents: \(y^{4} \times y^{7} = y^{4+7} = y^{11}\)
04

- Combine the Results

Combine the results from Step 2 and Step 3: \(15 y^{11}\)

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Key Concepts

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

Law of Exponents
The law of exponents is a fundamental rule in algebra that helps us handle terms with the same base. When you are multiplying terms with the same base, you simply add their exponents. For instance, if you have two terms like \( y^4 \) and \( y^7 \), you can combine them by adding the exponents: 4 and 7. This results in \( y^{4+7} \), which simplifies to \( y^{11} \).
This rule is crucial when working with polynomials, as it allows us to condense more complex expressions into simpler forms.
Simplifying Expressions
Simplifying expressions means making them easier to understand or work with, by performing operations step by step. Let's go through the simplification process:
  • First, identify similar terms or operations that can be combined or simplified.
  • Next, perform these operations according to the algebraic rules, such as the law of exponents.
  • Always follow the order of operations (PEMDAS/BODMAS) to ensure you simplify correctly.
In our example, we started with the expression \(5 y^{4}\times 3 y^{7}\). We first rewrote it for clarity and then multiplied the numerical coefficients and applied the law of exponents.
Combining these results gave us the simplified expression: \(15 y^{11}\). Simplifying expressions makes them easier to interpret and solve, which is particularly useful in more complex algebraic problems.
Numerical Coefficients
Numerical coefficients are the constant numbers that appear before the variable terms in an algebraic expression. In the expression \(5 y^4\), 5 is the numerical coefficient. When multiplying polynomials, it's important to handle these coefficients correctly:
  • Identify and separate the numerical coefficients from the variable parts.
  • Multiply the numerical coefficients together.
In the given exercise, we had the coefficients 5 and 3. By multiplying them, we got \(5 \times 3 = 15\).
This product becomes the new coefficient in the simplified expression.
Correctly managing these coefficients ensures that your final expression accurately represents the quantities involved.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}-1\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

If it costs \(\$ 15\) plus \(\$ 2\) per day to rent a chain saw, the binomial $$2 x+15$$ gives the cost in dollars to rent the chain saw for \(x\) days. Evaluate this binomial for \(x=6\). Use the result to fill in the blanks: If the saw is rented for_________ days, then the cost is _________ dollars.

Add or subtract as indicated. \(\left(3 k^{2} h^{3}+5 k h+6 k^{3} h^{2}\right)-\left(2 k^{2} h^{3}-9 k h+k^{3} h^{2}\right)\)

Graph each equation by completing the table of values. $$ \begin{aligned} &y=(x+3)^{2}\\\ &\begin{array}{c|c|c|c|c|c} x & -5 & -4 & -3 & -2 & -1 \\ \hline y & & & & & \end{array} \end{aligned} $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (b-10)(b+10) $$
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