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Chapter 5: Problem 72

. \(\left(-\frac{5 a^{3}}{6 b^{5}}\right)\left(\frac{2 a^{8}}{15 b^{4}}\right)\)

Short Answer

Expert verified
-\frac{a^{11}}{9 b^{9}}

Step by step solution

01

- Multiply the Numerators

Multiply the numerators of both fractions. This means multiplying \(-\frac{5 a^{3}}{6 b^{5}}\) by \(\frac{2 a^{8}}{15 b^{4}}\).\(-5 a^3 \times 2 a^8 = -10 a^{11}\).
02

- Multiply the Denominators

Next, multiply the denominators of both fractions. This means \(6 b^5 \times 15 b^4 = 90 b^9\).
03

- Combine and Simplify

Combine the results from steps 1 and 2 to create a new fraction: \(-\frac{10 a^{11}}{90 b^{9}}\). Then, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 10. This results in \(-\frac{a^{11}}{9 b^{9}}\).

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Key Concepts

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

Multiplying Fractions
Multiplying fractions involves a straightforward process. You multiply the numerators (the top numbers) and then the denominators (the bottom numbers). Let's break this down:

For example, if you have \(\frac{a}{b} \times \frac{c}{d}\), you multiply the numerators: \(a \times c\) and the denominators: \(b \times d\).

This yields a new fraction: \(\frac{ac}{bd}\).

In our exercise, we multiplied \(-\frac{5 a^{3}}{6 b^{5}}\) by \(\frac{2 a^{8}}{15 b^{4}}\).

First, multiply the numerators: \(-5 a^3 \times 2 a^8 = -10 a^{11}\).

Next, multiply the denominators: \(6 b^5 \times 15 b^4 = 90 b^9\).

Combining these results gives us: \(-\frac{10 a^{11}}{90 b^9}\).
Algebraic Expressions
Algebraic expressions contain variables, numbers, and operations. When dealing with variables in fractions, you need to treat them like regular numbers during multiplication.

For example, in the expression \(-5a^3\), \-5\ is the coefficient (the numerical part) and \a^3\ is the variable part raised to a power.

In our exercise, the expressions involved were \(-\frac{5 a^{3}}{6 b^{5}}\) and \(\frac{2 a^{8}}{15 b^{4}}\). While multiplying these algebraic expressions, we handled the variable parts just like the numerical ones, multiplying the bases and adding the exponents:

\(-5 a^3 \times 2 a^8 = -10 a^{11}\).

This rule applies to both the numerator and denominator.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. This means dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our exercise, the fraction obtained was \(-\frac{10 a^{11}}{90 b^9}\). First, find the GCD of 10 and 90, which is 10.

Next, divide both the numerator and the denominator by their GCD:

\(\frac{10 a^{11}}{10} = a^{11}\)

and

\(\frac{90 b^9}{10} = 9 b^9\).

This results in the simplified fraction: \(-\frac{a^{11}}{9 b^9}\).

Remember, simplifying fractions helps make them easier to work with and ensures they are in their most reduced form.

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Most popular questions from this chapter

\(\left(5 w^{4}-15 w^{2}+60 w+20\right) \div(5 w)\)

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The height of a triangle is \(6 \mathrm{~cm}\) less than the base. a. If \(b=\) base, write a polynomial expression in \(b\) that represents the height, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(b\) that represents the area.
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