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Chapter 6: Problem 432

Mixed Practice $$\frac{32 y^{5}}{8 y^{2}}-\frac{60 y^{10}}{5 y^{7}}$$

Short Answer

Expert verified
-8y^3

Step by step solution

01

Simplify the First Fraction

Simplify the first term, \[\frac{32 y^{5}}{8 y^{2}}\text{.}\] Divide the coefficients (32 and 8) and subtract the exponents of y: \(32 \div 8 = 4\) and \(y^{5} \div y^{2} = y^{5-2} = y^{3}\text{.}\) Therefore, \[\frac{32 y^{5}}{8 y^{2}} = 4 y^{3}\text{.}\]
02

Simplify the Second Fraction

Simplify the second term, \[\frac{60 y^{10}}{5 y^{7}}\text{.}\] Divide the coefficients (60 and 5) and subtract the exponents of y: \(60 \div 5 = 12\) and \(y^{10} \div y^{7} = y^{10-7} = y^{3}\text{.}\) Therefore, \[\frac{60 y^{10}}{5 y^{7}} = 12 y^{3}\text{.}\]
03

Calculate the Final Expression

Now, subtract the second term from the first term: \[\begin{aligned} 4 y^{3} - 12 y^{3}. \end{aligned}\] Factor out \(y^{3}\text{:}\) \[(4 - 12) y^{3} \text{.} \] Calculate \[(4 - 12)= -8\text{.} \] Therefore, the final expression is \[-8 y^{3}\text{.}\]

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. In the given exercise, we have algebraic fractions, which involve expressions divided by each other. To simplify these fractions, it鈥檚 essential to handle both the numerical coefficients and the variables appropriately.
  • A variable is a symbol, like y, that represents an unknown value.
  • Coefficients are the numbers in front of the variables, like 32 and 8 in the first fraction.
  • Operations like addition, subtraction, multiplication, and division are used to manipulate these expressions.
Understanding how to manage these components is crucial for solving algebraic expressions.
Using Exponent Rules
Exponent rules help us simplify expressions where variables are raised to powers. Here are the core rules used in the problem:
  • Product of Powers: When multiplying two expressions with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: When dividing two expressions with the same base, subtract their exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
In the solution:
  • For the first fraction, \(\frac{32 y^{5}}{8 y^{2}}\), the exponents of y are 5 and 2. So, we subtract 2 from 5 to get \(y^{5-2} = y^{3}\).
  • For the second fraction, \(\frac{60 y^{10}}{5 y^{7}}\), we subtract 7 from 10, resulting in \(y^{10-7} = y^{3}\).
Understanding these rules is key to simplifying expressions efficiently.
Dividing Coefficients
Coefficient division is an important part of simplifying fractions in algebra. Coefficients are the numerical parts of terms that have variables attached. For the problem stated:
  • First, in \(\frac{32 y^{5}}{8 y^{2}}\), we divide 32 by 8 to get 4.
  • Next, in \(\frac{60 y^{10}}{5 y^{7}}\), we divide 60 by 5 to get 12.
After dividing these coefficients, we can further simplify by handling the exponents. Combining all these steps helps break down the problem into manageable parts. This method makes it easier to follow and solve complex expressions.

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Most popular questions from this chapter

Divide the monomials. $$\frac{\left(10 m^{5} n^{4}\right)\left(5 m^{3} n^{6}\right)}{25 m^{7} n^{5}}$$

Divide each polynomial by the binomial. $$\left(3 b^{3}+b^{2}+2\right) \div(b+1)$$

Mixed Practice (a) \(24 a^{5}+2 a^{5}\) (b) \(24 a^{5}-2 a^{5}\) (c) \(24 a^{5} \cdot 2 a^{5}\) (d) \(24 a^{5} \div 2 a^{5}\)

Simplify. (a) \(4^{-2}\) (b) \(10^{-3}\)

Divide the monomials. $$\frac{64 q^{11} r^{9} s^{3}}{48 q^{6} r^{8} s^{5}}$$
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