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Chapter 5: Problem 11

Find each product. $$ y^{5} \cdot 9 y \cdot y^{4} $$

Short Answer

Expert verified
9y^{10}

Step by step solution

01

- Combine Like Terms

Identify and combine all the terms with the same base. In this case, the base is 'y': \(y^5, y, y^4\)
02

- Use the Product of Powers Property

Use the property of exponents that states when multiplying like bases, you add the exponents: \(y^5 \times y \times y^4 = y^{5+1+4}\)
03

- Simplify the Exponents

Add the exponents together:\(y^{5+1+4} = y^{10}\)
04

- Simplify the Expression with Constants

Multiply the constant 9 with the simplified expression: \(9 \times y^{10}\)

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Key Concepts

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

Product of Powers
When multiplying terms that have the same base, we can use the 'product of powers' property. This property tells us that we simply add the exponents of the terms being multiplied. For example, if we have two terms, say \(a^m\) and \(a^n\), and we multiply them together, we combine them by adding their exponents: \ \(a^m \times a^n = a^{m+n}\). So in our exercise, we apply this rule to the terms with the base 'y'. \
\ The original exercise has the terms: \
    \
  • 袄(测镑5袄蹿补尘颈濒械谤测镑4袄)
  • \(y\famil械y^4\)(which is equivalent to \(y^1\))
    • \
Find the sum of the exponents: \ \(y^5 \times y \times y^4 = y^{5+1+4} = y^{10}\). Now, 'y' is raised to the power of 10. \ This combines like terms with the 'product of powers' property beautifully.
Combining Like Terms
In algebra, it's essential to combine like terms to simplify expressions. Like terms are terms that have the same variables raised to the same power. For example, \(2x\) and \(3x\) are like terms, but \(2x\) and \(2y\) are not. \
\ In our problem, the like terms are: \
    \
  • \( \text{y-terms}: y^5, y, y^4 \)
    • \
  • \(9\famil褍 \text{constant term, not a like term with y-terms}\)
    • \
By recognizing that all these terms have the same base 'y', we can combine them by adding their exponents, as described in the previous section. Once we combine the exponents, we focus on the constants. The constant '9' remains the same, simply multiplied to the entire combined term: \ \(y^5 \times y \times y^4\), which gave us \ \(y^{10}\). Now multiply it by 9 to get the final result: \(9 \times y^{10} \).
Simplifying Expressions
The last step is to simplify the entire expression. Simplifying is crucial to making a problem easier to read and solve. \
With our example, after combining like terms and applying the product of powers, we arrived at \(y^{10}\).\ Now, the expression includes a constant. We need to incorporate it to finalize our simplification. \ So, we have:\(y^{5}\times9y\times y^{4}\rightarrowy^{10}\times 9 \). And the simplified form is: \ \(9y^{10}\)
\ Always remember to look for opportunities to simplify at every step. Combining like terms and applying exponent rules early makes these expressions much easier to handle.

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