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Chapter 10: Problem 537

Simplify each expression. $$(4 y)^{2}(8 y)$$

Short Answer

Expert verified
128y^3

Step by step solution

01

- Apply the Power Rule

Simplify \( (4y)^2 \) by applying the power rule \( (a^m)^n = a^{m \cdot n}. \) Therefore, \( (4y)^2 = 4^2 \cdot y^2 = 16y^2. \)
02

- Multiply the Expressions

Multiply \( 16y^2 \) by \( 8y. \) When you multiply these together, you get \( 16y^2 \cdot 8y = 128y^3. \)

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Key Concepts

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

Power Rule in Algebra
The Power Rule in algebra is a fundamental tool that helps simplify expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents. For example, if you have \((a^m)^n= a^{m \cdot n}\).\
\
In the given exercise, we start by applying the power rule to \((4y)^2)\). This means we must square both 4 and y.\
\
Step-by-step: \
    \
  • Square the number 4: \(4^2 = 16\).\
  • Square the variable y: \(y^2\).\
      \
      Thus, \((4y)^2 = 16y^2\). This simplifies the original part of the expression effectively.
Multiplying Monomials
Multiplying monomials involves combining coefficients and adding the exponents of like variables.\
\
In the next step of our solution, we multiply \(16y^2\) by \(8y\).\
\
Here鈥檚 how to break it down: \
    \
  • Multiply the coefficients (numbers): \(16 \cdot 8 = 128\).\
  • Add the exponents for the variable y: \(y^2 \cdot y = y^{2+1} = y^3\).\
      \
      So, when you multiply \(16y^2\) and \(8y\), you get \(128y^3\). This approach makes simplifying expressions systematic and straightforward.
Exponents
Exponents indicate how many times a number, known as the base, is multiplied by itself. They appear frequently in algebraic expressions and follow specific rules.\
\
Let鈥檚 recap some of the essential rules for working with exponents used in this exercise: \
    \
  • Power Rule: \( (a^m)^n = a^{m \cdot n} \).\
  • Multiplication Rule: \(a^m \cdot a^n = a^{m+n}\).\
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is 1: \(a^0 = 1\).\
      \
      These rules help simplify complex algebraic expressions efficiently. For example, in our exercise, the multiplication rule helped us combine \(y^2 \) and \(y\) into \(y^3\). Using such rules can significantly ease the process of simplifying algebraic expressions.

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