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Chapter 5: Problem 87

Multiply. $$ \left(\frac{1}{2} a\right)\left(4 a^{4}\right)\left(a^{5}\right) $$

Short Answer

Expert verified
The product is \(2a^{10}\).

Step by step solution

01

Organize the multiplication

First, let's organize the terms and understand what we need to multiply: \[\left(\frac{1}{2} a\right)\left(4 a^{4}\right)\left(a^{5}\right) \] This expression involves multiplying coefficients (\(\frac{1}{2}\) and \(4\)) and combining powers of \(a\).
02

Multiply the coefficients

Multiply the numerical coefficients \(\frac{1}{2}\) and \(4\):\[\frac{1}{2} \times 4 = 2\] So, the result of multiplying the coefficients is \(2\).
03

Combine like terms with the same base

Next, multiply the powers of \(a\). When multiplying terms with the same base, add the exponents:\[a^{1} \times a^{4} \times a^{5}\]Add the exponents: \[1 + 4 + 5 = 10\]Thus, the power of \(a\) in the final expression is \(a^{10}\).
04

Write the final expression

Combine the coefficient obtained from Step 2 with the power of \(a\) obtained in Step 3 to form the final expression:\[2a^{10}\]

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Key Concepts

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

Understanding Coefficients
In polynomial multiplication, coefficients are the numerical factors that multiply the variables. In our specific problem, the coefficients are the numbers that multiply the variable \( a \). Here, we start with coefficients \( \frac{1}{2} \) and \( 4 \). To find the resultant coefficient, you simply multiply these numbers together.
  • Identify the numerical parts: \( \frac{1}{2} \) and \( 4 \).
  • Perform the multiplication: \( \frac{1}{2} \times 4 = 2 \).
  • Use this result as the numerical part of your final expression.
Understanding coefficients is crucial because they determine the size of the term in your polynomial. In any term, think of them as the scaling factor of the expression.
Exploring Exponents
Exponents represent the number of times a base is multiplied by itself. For polynomials, especially when multiplying terms, managing exponents is key to simplifying the expression correctly.When multiplying terms with the same base, you add the exponents together. In the expression \( a \times a^4 \times a^5 \), each \( a \) has its own exponent:
  • First coefficient has an implied exponent of \( 1 \) (\( a^1 \)).
  • The second term is \( a^4 \).
  • The third term is \( a^5 \).
Adding these exponents together, we find:\[1 + 4 + 5 = 10\]The result is \( a^{10} \), meaning \( a \) is effectively multiplied by itself ten times in the product.
Identifying Like Terms
Like terms in polynomials are those terms that have the same variable raised to the same power. When multiplying polynomials, it's essential to recognize and properly combine like terms to simplify expressions accurately.In our example:
  • Every term has the variable \( a \).
  • Each term's \( a \) is raised to a different power.
Since each term shares the base \( a \), they are "like terms" in the sense that they can all be combined in multiplication through exponent addition, even if the exponents differ. Properly identifying them allows for the seamless simplification into a result of \( 2a^{10} \). This simplification helps to maintain the polynomial's integrity and makes further operations easier.

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Most popular questions from this chapter

Perform each division. $$ \frac{-18 w^{6}-9}{9 w^{4}} $$

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