Problem 16 Multiply.$$\left(3 a^{3} b^{6}\r... [FREE SOLUTION]

Chapter 10: Problem 16

Multiply.$$\left(3 a^{3} b^{6}\right)\left(12 a^{2} b^{9}\right)$$

Short Answer

Expert verified

The product is $ 36 a^{5} b^{15} $.

Step by step solution

Multiply the Coefficients

The coefficients of the two expressions are 3 and 12. To find the product, multiply these coefficients together:\[ 3 \times 12 = 36 \]

Add the Exponents of the Same Base for 'a'

The base 'a' has exponents 3 and 2 in the expressions. To multiply terms with the same base, add the exponents:\[ a^{3 + 2} = a^{5} \]

Add the Exponents of the Same Base for 'b'

Similarly, the base 'b' has exponents 6 and 9. Add these exponents:\[ b^{6 + 9} = b^{15} \]

Combine the Results

Combine the results from the previous steps to get the product:\[ 36 a^{5} b^{15} \]

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Key Concepts

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

Coefficients Multiplication

When multiplying polynomials, one of the first steps involves coefficients multiplication. The coefficient is the numerical factor in a term. For example, in the term $3a^3b^6$, 3 is the coefficient. To multiply two terms, simply multiply their coefficients together. This is a straightforward arithmetic operation.
Take the coefficients from each expression 鈥� in this case, 3 from $3a^3b^6$ and 12 from $12a^2b^9$ 鈥� and multiply them:

Step: Multiply $3 \times 12$
Result: $36$

This coefficient multiplication gives you the numerical factor for the product of the two expressions. It's important to do this step accurately as it impacts the whole expression's correctness.

Exponents Addition

In polynomial multiplication, when you have terms with the same base, you add the exponents. This is a fundamental rule of exponents that applies to numbers and variables alike. For instance, with the base 'a' in our example, the term $a^3$ is to be multiplied by $a^2$. Adding their exponents gives you:

Step: Add the exponents $3 + 2$ for the base 'a'
Result: $a^5$

Similarly, for the base 'b', take the exponents 6 and 9:

Step: Add the exponents $6 + 9$ for the base 'b'
Result: $b^{15}$

This addition of exponents ensures the powers of each base are correctly combined in the product of the terms, maintaining consistency with mathematical rules.

Product of Variables with Exponents

The product of variables with exponents involves several steps tying together the concept of coefficients multiplication and exponents addition. It's all about systematically applying these rules to combine the expressions accurately.
Start first with the multiplication of coefficients as described previously. Then move on to the identifiers. For the given problem:

The variable 'a': Combine the results by forming $a^{5}$, since $3 + 2 = 5$.
The variable 'b': Combine the results as $b^{15}$, since $6 + 9 = 15$.

Finally, compile these results together:

Overall product: $36a^5b^{15}$

This expression is the final result of multiplying the polynomials. Every step and rule applied here is crucial for maintaining the structural integrity of the resulting polynomial.

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91影视

Multiply.$$\left(3 a^{3} b^{6}\right)\left(12 a^{2} b^{9}\right)$$

Short Answer

Step by step solution

Multiply the Coefficients

Add the Exponents of the Same Base for 'a'

Add the Exponents of the Same Base for 'b'

Combine the Results

Key Concepts

Coefficients Multiplication

Exponents Addition

Product of Variables with Exponents

One App. One Place for Learning.

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