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Chapter 10: Problem 108

In the following exercises, simplify each expression. $$(2 y)^{3}(6 y)$$

Short Answer

Expert verified
48y^{4}

Step by step solution

01

Understand the expression

The expression given is \( (2y)^{3}(6y) \). The goal is to simplify it.
02

Expand the exponential term

Expand \( (2y)^{3} \). This equals \( 2^{3} \times y^{3} = 8y^{3} \).
03

Multiply the remaining terms

Now multiply \( 8y^{3} \) by \( 6y \). This results in \((8y^{3})(6y) = 8 \times 6 \times y^{3} \times y \).
04

Combine like terms

Combine the constants: \( 8 \times 6 = 48 \), and combine the powers of \ y \: \ y^{3} \times y = y^{4} \). The expression simplifies to \( 48y^{4} \.

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Key Concepts

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

exponents
Exponents are a shorthand way to represent repeated multiplication of the same number or variable. If you see an expression like o a^{n} this means 'a' is multiplied by itself 'n' times. For example, (2y)^{3} means we multiply (2y) by itself three times: (2y) 脳 (2y) 脳 (2y). When simplifying, you can break this down into coefficients and variables separately. So (2y)^{3} = 2^{3} 脳 y^{3} = 8y^{3} . Remember to treat each part separately: the number (2) and the variable (y).
multiplication of polynomials
When multiplying polynomials, you need to apply the distributive property: every term in one polynomial is multiplied by every term in the other. In our case: (8y^{3})(6y) we multiply the coefficients and the variables separately:
  • 8 (from 8y^{3}) times 6 (from 6y) gives us 48.
  • y^{3} (from 8y^{3}) times y (from 6y) gives us y^{4}.
Combining both results, we get: 48y^{4}.
combining like terms
Combining like terms means simplifying an expression by adding or subtracting terms with the same variable raised to the same power. For example, if you have y^{3} and y^{4} , you cannot combine them because they are not 'like terms'. Only terms with the same exponent can be added together. In as the final result of our given expression , 48y^{4} there are no like terms to combine, since the expression has been already simplified to its lowest form.

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Most popular questions from this chapter

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