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Chapter 4: Problem 114

Simplify. Assume no division by 0. $$\left(4 t^{3} t^{6} t^{2}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \(16 t^{22}\).

Step by step solution

01

Understand the problem

The given expression is \((4 t^{3} t^{6} t^{2})^{2}\). We need to simplify this.
02

Apply the rules of exponents

Since all the terms inside the parentheses are being multiplied, and they have the same base \(t\), we can combine the exponents by adding them. This gives \(4 t^{3+6+2} = 4 t^{11}\).
03

Apply the power of a power rule

We apply the rule which states that \((a b)^n = a^n b^n\). Here, \(a=4\), \(b=t^{11}\), and \(n=2\). Therefore, \((4 t^{11})^2 = 4^2 (t^{11})^2 = 16 t^{22}\).

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Key Concepts

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

Rules of Exponents
Exponent rules are essential when simplifying expressions involving powers. These rules help us manage expressions with similar bases efficiently. The two most common rules are:
  • Product of Powers Rule: When multiplying like bases, keep the base the same and add the exponents. For example, if you have two terms like \(a^m \times a^n\), the product will be \(a^{m+n}\).
  • Power of a Product Rule: This rule states that when you have a power outside the parenthesis affecting products inside, you can distribute the power to each term. For example, \((xy)^n = x^n y^n\).
When faced with an expression, always look to apply these rules. It greatly simplifies your work and reduces the chances of mistakes. Remember, consistent practice makes these rules intuitive.
Power of a Power Rule
The power of a power rule is a handy tool in exponentiation. It helps simplify expressions where a power is raised to another power. For instance, in the expression \((a^m)^n\), the powers combine by multiplying them, resulting in \(a^{mn}\).
This rule allows you to quickly shift between different power expressions and make complex simplifications manageable.
In the given expression \((4t^{11})^2\), the power of two is distributed as \(4^2 (t^{11})^2\).
Applying the power to the base \(t^{11}\) results in \(t^{22}\), demonstrating the effectiveness of understanding this rule.
Multiplying Exponents
Multiplying exponents seems tricky at first, but it's straightforward once you grasp the rules. When the bases of the exponents are the same, you add the exponents; when exponents multiply, you multiply the powers.
For example, consider the expression \(t^3 \times t^6 \times t^2\). Using the product of powers rule, you add the exponents: \(t^{3+6+2} = t^{11}\). This simplifies the expression considerably and reduces errors.
  • Remember: the base must be the same when adding or multiplying exponents for simplification.
  • If two different bases are involved, work with each separately before combining.
Practicing with different base expressions helps build confidence and mastery over simplifying exponent expressions efficiently.

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