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Chapter 1: Problem 25

Simplify. $$\left(3 y^{3}\right)^{4}\left(4 y^{2}\right)^{-3}$$

Short Answer

Expert verified
The simplified expression is \(\frac{81}{64}y^{6}\).

Step by step solution

01

Apply the Power of a Power Rule

The expression given is \((3y^3)^4(4y^2)^{-3}\). First, apply the power of a power rule: \((a^m)^n = a^{mn}\). This gives us: \((3y^3)^4 = 3^4 (y^3)^4 = 81y^{12}\) and \((4y^2)^{-3} = 4^{-3}(y^2)^{-3} = \frac{1}{64}y^{-6}\). So the expression simplifies to \(81y^{12} \times \frac{1}{64}y^{-6}\).
02

Multiply the Coefficients

Multiply the coefficients 81 and \(\frac{1}{64}\). This results in \(\frac{81}{64}\). So the expression becomes \(\frac{81}{64}y^{12}y^{-6}\).
03

Apply the Product of Powers Property

For the powers of \(y\), apply the product of powers property: \(y^a \times y^b = y^{a+b}\). Add the exponents \(12 + (-6) = 6\). This simplifies the expression to \(\frac{81}{64}y^6\).

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Key Concepts

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

Power of a Power Rule
The "Power of a Power Rule" is a fundamental concept in simplifying exponential expressions. When you raise an exponent to another power, you multiply the exponents. This rule can be written as
  • \((a^m)^n = a^{m\cdot n}\).
Let's break down the exercise using this rule to make it clearer.Consider the expression
  • \((3y^3)^4\)
Here, both 3 and \(y^3\) are raised to the power of 4. By applying the power of a power rule, we get
  • \(3^4 = 81\)
  • \((y^3)^4 = y^{3\cdot 4} = y^{12}\).
Thus, \((3y^3)^4\) simplifies to
  • \(81y^{12}\).
Similarly, for
  • \((4y^2)^{-3}\),
applying the rule gives us
  • \(4^{-3} = \frac{1}{64}\)
  • \((y^2)^{-3} = y^{2\cdot (-3)} = y^{-6}\).
This simplifies to
  • \(\frac{1}{64}y^{-6}\).
With these simplifications, the original expression becomes
  • \(81y^{12} \times \frac{1}{64}y^{-6}\).
Product of Powers Property
The "Product of Powers Property" helps us simplify expressions with the same base. This property states that when you multiply like bases, you add the exponents.In math form, that's written as
  • \(a^m \times a^n = a^{m+n}\).
In our example, after applying the power of a power rule, we have the expression
  • \(81y^{12} \times \frac{1}{64}y^{-6}\).
Here's how we use the product of powers property:
  • For the coefficients, multiply \(81\) by \(\frac{1}{64}\), resulting in \(\frac{81}{64}\).
  • For the variable part, apply the property \(y^{12}\) and \(y^{-6}\). Add the exponents: \(12 + (-6) = 6\).
So, the expression with the combined powers is
  • \(\frac{81}{64}y^6\).
By adding the exponents of \(y\), we've effectively simplified the expression!
Simplifying Expressions
"Simplifying Expressions" means to reduce them into a form that's easier to work with or understand, without changing their value. Let鈥檚 piece together the steps we've covered to ensure you've grasped the simplification process.
  • First, apply the power of a power rule to distribute the exponent to each base within the parentheses, simplifying terms like
    • \((3y^3)^4\)
    to
    • \(81y^{12}\)
    and
    • \((4y^2)^{-3}\)
    to
    • \(\frac{1}{64}y^{-6}\).
  • Next, involve the product of powers property. Multiply coefficients and add any corresponding exponents:
    • \(81 \times \frac{1}{64} = \frac{81}{64}\)
    • For the base \(y\), combine exponents: \(y^{12 + (-6)} = y^6\).
  • Putting it all together, we've simplified the entire expression to:
    • \(\frac{81}{64}y^6\).
Simplifying makes the expression concise and easier to interpret. With practice, identifying which rules to apply becomes second nature.

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