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Chapter 9: Problem 18

Multiply the following expressions. \(2 y^{4} \cdot 7 y^{4}\)

Short Answer

Expert verified
The product is \(14y^8\).

Step by step solution

01

Multiply the Coefficients

First, identify and multiply the numerical coefficients (constants) in each expression. The coefficients in the given expressions are 2 and 7. Multiply these numbers together: \[2 \times 7 = 14\]
02

Apply the Power of a Power Rule

When multiplying terms with the same base, you add the exponents. Here, you have \(y^4\) in both expressions:\[y^4 \cdot y^4 = y^{4+4} = y^8\]
03

Combine the Results

Combine the results from Step 1 and Step 2 to form the final expression. From Step 1, we have the coefficient 14, and from Step 2, we have the term \(y^8\). Therefore, the product of the expressions is:\[14y^8\]

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Key Concepts

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

Coefficients
Coefficients are the numerical factors in a term or expression. They multiply the variable part of the expression. In the example of multiplying expressions \(2y^4 \cdot 7y^4\), the coefficients are the numbers 2 and 7. These are the numerical values you multiply first before you work with the variables.

Here's how it works:
  • Identify the coefficients: 2 and 7.
  • Multiply them together: \(2 \times 7 = 14\).
This process gives you the new coefficient for the product. This first step focuses solely on numbers, so it simplifies the expression and sets the base for further calculations.
Power of a Power Rule
The power of a power rule is crucial when you're dealing with variables that have exponents, especially when multiplying them. When you see an expression like \((y^4)\), and it's multiplied with another \((y^4)\), you are using the power of a power rule. This rule tells us that when multiplying like bases, you should add their exponents together.

In formulas, the rule can be described as:
  • If you have \(x^a \cdot x^b\), the result is \(x^{a+b}\).
Applying this rule to our example, \(y^4 \cdot y^4 = y^{4+4} = y^8\). We've added the exponents because they share the same base, \(y\). By understanding and using this rule, we can quickly simplify expressions involving powers.
Exponents
Exponents are the tiny numbers written above and to the right of a base number or variable. They indicate how many times the base is multiplied by itself. In the expression \(y^4\), 4 is the exponent. It means \(y\) is being multiplied by itself four times: \(y \cdot y \cdot y \cdot y\).

When multiplying exponents with the same base, the exponents are added, as per the power of a power rule. This method simplifies complex expressions easily by reducing them to a single term. Here鈥檚 the takeaway:
  • Exponents signal repeated multiplication of the base.
  • Adding exponents is crucial when multiplying like terms.
  • They help in simplifying and solving expressions.
Understanding how to manage exponents correctly allows us to multiply expressions efficiently and accurately.

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Most popular questions from this chapter

Simplify the following expressions. \(\left(9 x^{4}\right)^{2}\)

Expand and multiply. $$(2 x+3)^{2}$$

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(3.5 \times 10^{-4}\right)\left(4.2 \times 10^{5}\right)}{7 \times 10^{3}}$$

Simplify the following expressions. \(\left(a^{2} b^{4}\right)^{3}\left(a b^{5}\right)^{4}\left(a^{4} b\right)^{2}\)

Multiply the following expressions. $$(3 x y-10) 2 x y$$
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