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Chapter 6: Problem 142

Multiply the monomials. $$\left(\frac{5}{8} x^{3} y\right)\left(24 x^{5} y\right)$$

Short Answer

Expert verified
15x^8 y^2

Step by step solution

01

Multiply the Coefficients

First, identify and multiply the coefficients of each monomial. The coefficients are \(\frac{5}{8}\) and 24. Compute the product: \[ \frac{5}{8} \times 24 = 15 \]
02

Multiply the Variables

Next, multiply the variables with the same base by adding their exponents. For \(x\): \[ x^{3} \times x^{5} = x^{3+5} = x^{8} \] For \(y\): \[ y \times y = y^{1+1} = y^{2} \]
03

Combine the Results

Combine the results from the previous steps to obtain the final expression: \[ 15x^8 y^2 \]

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Key Concepts

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

Coefficients Multiplication
When multiplying monomials, the first step is to multiply the coefficients. Coefficients are the numerical parts of the terms.
In our example, the coefficients are \(\frac{5}{8}\) and 24.
To multiply these coefficients, follow these steps:
  • Multiply the numerators and denominators separately: \[ \frac{5}{8} \times 24 = \frac{5 \times 24}{8} \]
  • Simplify the fraction: \[ \frac{120}{8} = 15 \]
So, the product of the coefficients is 15.
This is a straightforward process and can be done by hand or with a calculator.
Exponent Addition
Next, we need to multiply the variables. When multiplying variables with the same base, we add their exponents.
This is a key rule in algebra called exponent addition.
Let鈥檚 look at the variables in our example:
  • For the x-terms: \[ x^{3} \times x^{5} = x^{3+5} = x^{8} \]
  • For the y-terms: \[ y \times y = y^{1+1} = y^{2} \]
You add the exponents because of the properties of exponents.
This makes combining variables easy and systematic.
Algebraic Expressions
Now, we combine both the coefficients and the variables with their new exponents.
This gives the final algebraic expression. Here鈥檚 how we put it all together:
  • We have the coefficient from step 1, which is 15.
  • We have the product of the x-terms: \[ x^{8} \]
  • We have the product of the y-terms: \[ y^{2} \]
Combine these parts to get the final result:
\[ 15x^8 y^2 \].
Understanding how to perform these operations systematically helps in managing more complex algebraic tasks. It's like building a foundation for advanced algebra concepts.

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Most popular questions from this chapter

Simplify. (a) \(2^{-8}\) (b) \(10^{-2}\)

Divide each polynomial by the binomial. $$\left(4 x^{2}-17 x-15\right) \div(x-5)$$

Simplify. (a) \(2 \cdot 5^{-1}\) (b) \((2 \cdot 5)^{-1}\)

Divide \(\frac{10 x^{2}+x-12}{2 x}\) and explain with words how you get each term of the quotient.

When Drake simplified \(-3^{0}\) and \((-3)^{0}\) he got the same answer. Explain how using the Order of Operations correctly gives different answers.
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