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Chapter 8: Problem 2

Perform each multiplication. $$ \frac{a^{3}}{b^{2} c^{2}} \cdot \frac{c^{5}}{a^{5}} $$

Short Answer

Expert verified
Question: Simplify the expression of the given multiplication: $$\frac{a^{3}}{b^{2} c^{2}} \cdot \frac{c^{5}}{a^{5}}$$ Answer: The simplified expression is $$\frac{c^{3}}{a^{2}b^{2}}$$.

Step by step solution

01

Multiply the fractions

Multiply the two fractions, combining the numerators and denominators: $$ \frac{a^{3}}{b^{2} c^{2}} \cdot \frac{c^{5}}{a^{5}} = \frac{a^{3}c^{5}}{b^{2}c^{2}a^{5}} $$
02

Simplify the expression using exponent rules

Use the exponent rules to simplify the expression. When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator: $$ a^{3} 梅 a^{5} = a^{(3-5)} = a^{-2} $$ and $$ c^{5} 梅 c^{2} = c^{(5-2)} = c^{3} $$
03

Substitute simplified expression

Replace each variable in the fraction with its simplified form: $$ \frac{a^{-2}c^{3}}{b^{2}} $$
04

Final Simplification

Rewrite the expression with positive exponents by moving the term with a negative exponent to the denominator: $$ \frac{c^{3}}{a^{2}b^{2}} $$ So, the simplified expression of the given multiplication is: $$ \frac{c^{3}}{a^{2}b^{2}} $$

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Key Concepts

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

Exponent Rules
Exponent rules are essential when working with algebraic expressions. They help to simplify and manipulate expressions by following specific guidelines. A key rule to remember is "When dividing like bases, subtract the exponents".
For example, when you have an expression like \(a^3 \div a^5\), subtract the exponents:
  • Write the first base with its exponent: \(a^3\)
  • Subtract the exponent of the denominator from that of the numerator: \(3 - 5 = -2\)
  • Result is \(a^{-2}\)
This rule works similarly with other bases, such as in the expression \(c^5 \div c^2\). Subtraction gives \(c^{5-2} = c^3\). Keep in mind:
  • Only subtract exponents when the bases are the same.
  • If the subtraction results in a negative exponent, it means reciprocal or invert the base under positive exponent rules.
Understanding these rules allows for simplification of complex expressions efficiently.
Fraction Multiplication
Fraction multiplication involves multiplying two fractions by multiplying their numerators and denominators separately. For example, for the fractions \( \frac{a^3}{b^2 c^2} \) and \( \frac{c^5}{a^5} \):
  • Multiply the numerators: \( a^3 \cdot c^5 = a^3 c^5 \)
  • Multiply the denominators: \( b^2 c^2 \cdot a^5 = b^2 c^2 a^5 \)
  • The resulting fraction is: \( \frac{a^3 c^5}{b^2 c^2 a^5} \)
It is as simple as basic multiplication of numbers, but you need to be attentive to the algebraic expressions you are working with. Avoid common mistakes like improperly combining unlike terms or forgetting to multiply the entire numerator by the entire denominator. Once multiplied, the expression can often be simplified further using exponent rules.
Simplifying Expressions
Simplifying expressions is the process of making an expression more manageable or clean by using mathematical methods. In algebra, these methods include using exponent rules, combining like terms, and simplifying fractions.
Consider the fraction \( \frac{a^{-2}c^{3}}{b^{2}} \). There's a negative exponent involved which we make positive by moving the base to the opposite part of the fraction:
  • Move \(a^{-2}\) from numerator to denominator to become \(a^2\).
  • The simplified expression becomes \( \frac{c^3}{a^2 b^2} \).
Simplifying expressions makes calculations and understanding of problems easier by presenting them in their simplest form. Always aim to remove negative exponents, reduce fractions, and ensure each term is in its simplest algebraic form for clarity and ease of use.

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Most popular questions from this chapter

For the following problems, perform the divisions. $$ \frac{20 y^{2}+15 y-4}{4 y+3} $$

When the same number is subtracted from both the numerator and denominator of \(\frac{3}{10},\) the result is \(\frac{1}{8}\). What is the number that is subtracted?

For the following problems, perform the divisions. $$ \left(6 a^{4}-2 a^{3}-3 a^{2}+a+4\right) \div(3 a-1) $$

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For the following problems, find the solution. Debbie can complete an algebra assignment in \(\frac{3}{4}\) of an hour. Sandi, who plays her radio while working, can complete the same assignment in \(1 \frac{1}{4}\) hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?
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