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Chapter 5: Problem 64

Find each product. \(-4 k^{2}\left(-3 k^{5} t^{3}\right)\)

Short Answer

Expert verified
12 k^{7} t^{3}

Step by step solution

01

Distribute the Constants

Multiply the constants \(-4\) and \(-3\). The product of two negative numbers is positive, so \(-4 \times -3 = 12\).
02

Apply the Law of Exponents for k

Add the exponents of \(k\). The exponents are \2\ and \5\, so \k^{2} \times k^{5} = k^{(2+5)} = k^{7}\.
03

Combine the Remaining Factors

Write down the product from Step 1 and multiply it by the result from Step 2, then include \(t^{3}\): \12 k^{7} t^{3}\.

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Key Concepts

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

Law of Exponents
When multiplying polynomials, understanding the law of exponents is crucial. This law dictates how to handle exponents when you multiply like bases. For instance, if you have two terms like \(k^2\) and \(k^5\), you need to add the exponents together to get the new exponent. So, \(k^2 \times k^5\) becomes \(k^{(2 + 5)} = k^7\). Remember, this rule only applies when the bases are the same. For any bases \(a\) and \(b\), \(a^m \times a^n = a^{(m+n)}\). This principle simplifies complex expressions and is a fundamental skill in algebra. Breaking it down like this can make even the toughest polynomial problems seem manageable.
Distributive Property
The distributive property is another key concept in solving polynomial multiplication. This property tells us that for any three numbers or variables, \(a(b + c) = ab + ac\). In the given problem, we distribute the constants first: \(-4\) and \(-3\). When we multiply them, we get \(12\) because multiplying two negative numbers results in a positive number. Next, we distribute this product to the variables \(k^2\) and \(k^5 t^3\). The distributive property is handy for breaking down more complicated expressions into simpler parts.
Combining Like Terms
Combining like terms is the final step in our polynomial multiplication process. Once you've applied both the law of exponents and the distributive property, you'll need to combine what's left. In our example, after distributing and applying the exponents, we're left with \(12k^7t^3\). Here, we only have one term left, but in more complicated problems, you might have to combine several like terms. Like terms are those that have the same variables raised to the same power. For example, \(3k^2 + 5k^2 = 8k^2\). Combining like terms simplifies your expression and makes it easier to understand the final result.

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Most popular questions from this chapter

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(\frac{3 k^{-2}}{k^{4}}\right)^{-1} \cdot \frac{2}{k} $$

Write each number in standard notation. $$ 2.54 \times 10^{-3} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{4 a^{5}\left(a^{-1}\right)^{3}}{\left(a^{-2}\right)^{-2}} $$

Simplify each expression. $$ 3(5+q)-2(6-q) $$

Find each product. \(-3 b^{5}\left(2 a^{3} b^{4}\right)\)
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