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Chapter 5: Problem 33

Multiply. $$ \frac{2}{3} a^{4}\left(6 a^{5}-12 a^{3}-\frac{5}{8}\right) $$

Short Answer

Expert verified
\( 4 a^{9} - 8 a^{7} - \frac{5}{12} a^{4} \)

Step by step solution

01

- Distribute the Fraction

Multiply \({\frac{2}{3}} a^{4}\) to each term inside the parentheses \((6 a^{5}-12 a^{3}-\frac{5}{8})\). This means you need to separately multiply \({\frac{2}{3}} a^{4}\) with \({6 a^{5}}\), \({-12 a^{3}}\), and \({-\frac{5}{8}}\).
02

- Multiply with the First Term

Calculate \({\frac{2}{3} a^{4} \cdot 6 a^{5}}\): \[ \frac{2}{3} a^{4} \cdot 6 a^{5} = \frac{2 \cdot 6}{3} a^{4} a^{5} = 4 a^{9} \]
03

- Multiply with the Second Term

Calculate \({\frac{2}{3} a^{4} \cdot -12 a^{3}}\): \[ \frac{2}{3} a^{4} \cdot -12 a^{3} = \frac{2 \cdot -12}{3} a^{4} a^{3} = -8 a^{7} \]
04

- Multiply with the Third Term

Calculate \({\frac{2}{3} a^{4} \cdot -\frac{5}{8}}\): \[ \frac{2}{3} a^{4} \cdot -\frac{5}{8} = \frac{2 \cdot -5}{3 \cdot 8} a^{4} = -\frac{10}{24} a^{4} = -\frac{5}{12} a^{4} \]
05

- Combine All the Terms

Put all the calculated terms together to form the final expression: \[ 4 a^{9} - 8 a^{7} - \frac{5}{12} a^{4} \]

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that allows us to multiply a term outside the parentheses by each term within the parentheses. When you have an expression like \(\frac{2}{3} a^{4}\big(6 a^{5}-12 a^{3}-\frac{5}{8}\big)\), you distribute \(\frac{2}{3} a^{4}\) to each term inside the parentheses. This means that you multiply \(\frac{2}{3} a^{4}\) with \(\big(6 a^{5}\big)\), \(\big(-12 a^{3}\big)\), and \(\big(-\frac{5}{8}\big)\) separately.
Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. They do not include division by a variable. In our example, terms like \(6 a^{5} - 12 a^{3} - \frac{5}{8}\) are considered polynomials. A polynomial can have one or more terms, and each term in a polynomial is called a monomial.
Exponents
Exponents are used to express repeated multiplication of a number by itself. For instance, \(a^{4}\) means \(a \cdot a \cdot a \cdot a\). When multiplying terms with the same base, you add the exponents: \(a^{4} \cdot a^{5} = a^{4+5} = a^{9}\). This rule is essential when you distribute and multiply polynomials, as shown in the exercise.
Fractions
Fractions represent part of a whole and are composed of a numerator and a denominator. Multiplying fractions involves multiplying the numerators together and the denominators together. In the given problem, when you distribute \(\frac{2}{3} a^{4}\), you multiply it with other fractions as well, necessitating an understanding of fraction multiplication. For instance, \(\frac{2}{3} \cdot \frac{5}{8} = \frac{2 \cdot 5}{3 \cdot 8} = \frac{10}{24}\), which simplifies to \(\frac{5}{12}\).

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Most popular questions from this chapter

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{5 x^{7} y}{-2 z^{4}}\right)^{3} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{-4 p^{8}}{3 m^{2} n^{3}}\right)^{3} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{x^{5}}{y^{2}}\right)^{7} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -16+5 $$

Review factoring expressions and solving equations. $$ 3 x-1=0 $$
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