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Chapter 0: Problem 39

\(29-46\) Simplify each expression. $$ \left(2 y^{2}\right)^{3} $$

Short Answer

Expert verified
\((2y^2)^3 = 8y^6\).

Step by step solution

01

Understand the Expression

The expression we need to simplify is \((2y^2)^3\). This involves an exponential term where a term inside the parentheses is raised to the power of 3.
02

Apply the Power of a Product Property

To simplify \((2y^2)^3\), use the "power of a product" rule, which states that \((ab)^n = a^n \cdot b^n\). Here, \(a = 2\) and \(b = y^2\), so \((2y^2)^3 = 2^3 \cdot (y^2)^3\).
03

Simplify Each Component

Calculate \(2^3\) which is 8. For \((y^2)^3\), use the "power of a power" rule, which states \((a^m)^n = a^{m \cdot n}\). So, \((y^2)^3 = y^{2 \cdot 3} = y^6\).
04

Combine the Simplified Terms

Now combine the terms from the previous step: \(2^3 \cdot (y^2)^3 = 8 \cdot y^6\). So, the expression \((2y^2)^3\) simplifies to \(8y^6\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of numbers, letters (variables), operation symbols, and sometimes parentheses. They represent mathematical concepts and can be simplified or solved. In our example,
  • The expression \((2y^2)^3\) is an algebraic expression.
  • It contains numbers, which are constants like 2, and a variable, \(y^2\), which changes its value.
  • The goal is often to simplify these expressions.
Algebraic expressions are the building blocks of algebra, allowing us to write general formulas and solve problems.Learning to recognize and work with these expressions is a crucial first step in mastering mathematics.
Power of a Power Rule
The power of a power rule is a fundamental concept in exponentiation. It helps simplify expressions where an exponential term is raised to another power. This rule can be expressed as:
  • \((a^m)^n = a^{m \cdot n}\),
where \(a\) is the base, and \(m\) and \(n\) are the exponents. In our exercise:
  • The term \((y^2)^3\) uses this rule.
  • Multiply the exponents 2 and 3 to get \(y^6\).
This rule simplifies the calculation process by combining exponents, helping manage large numbers efficiently.
Power of a Product Rule
The power of a product rule is another essential property used in exponentiation. It simplifies expressions where a product of factors is raised to a power. According to this rule:
  • \((ab)^n = a^n \cdot b^n\),
where \(a\) and \(b\) are factors and \(n\) is the common exponent. In the example given:
  • \((2y^2)^3\) involves multiplying both the number 2 and the term \(y^2\) raised to the power of 3.
  • The expression simplifies as follows: \(2^3 \cdot (y^2)^3\).
  • This gives us \(8 \cdot y^6\).
Recognizing and applying the power of a product rule helps you to efficiently break down complex expressions.

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Most popular questions from this chapter

Perform the multiplication or division and simplify. $$ \frac{x^{2}+7 x+12}{x^{2}+3 x+2} \cdot \frac{x^{2}+5 x+6}{x^{2}+6 x+9} $$

Perform the addition or subtraction and simplify. $$ \frac{1}{x^{2}}+\frac{1}{x^{2}+x} $$

Perform the multiplication or division and simplify. $$ \frac{t-3}{t^{2}+9} \cdot \frac{t+3}{t^{2}-9} $$

Perform the multiplication or division and simplify. \(\frac{x}{y / z}\)

Simplify the rational expression. $$ \frac{81 x^{3}}{18 x} $$
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