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Chapter 5: Problem 38

Simplify each expression. $$ \left(9 t^{8}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( 81t^{16} \).

Step by step solution

01

Understand the Problem

The given expression is \( (9t^8)^2 \). Our task is to simplify this expression. This involves applying the properties of exponents.
02

Apply the Power of a Product Rule

The expression can be written as \( (9)^2 \times (t^8)^2 \). This is because the rule \( (ab)^m = a^m \times b^m \) allows us to distribute the power of 2 over the entire product inside the parenthesis.
03

Calculate Each Power Separately

Compute \( 9^2 \) which is equal to 81. Then, compute \((t^8)^2\), applying the power of a power rule. This states that \((a^m)^n = a^{m \times n}\). Thus, \((t^8)^2 = t^{16}\).
04

Combine the Results

Combine the results from Step 3 to form the simplified expression: \( 81 \times t^{16} \). Therefore, the simplification of the original expression \( (9t^8)^2 \) is \( 81t^{16} \).

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Key Concepts

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

Power of a Product Rule
The power of a product rule is a key principle when working with exponents. It tells us how to distribute an exponent over a product inside parentheses. When you have an expression in the form \[ (ab)^m \]where "a" and "b" are any numbers or variables, and "m" is a positive integer,
  • The exponent "m" is applied to both "a" and "b" separately.
This means it becomes:\[ a^m \times b^m \]For example, applying this rule to \[ (9t^8)^2 \] we distribute the 2 over both 9 and \(t^8\), resulting in\[ 9^2 \times (t^8)^2 \].Understanding this rule can greatly simplify your calculations and make it easier to manage expressions where multiple terms are raised to a power.
Power of a Power Rule
The power of a power rule simplifies expressions where you have an exponent raised to another exponent, such as \[ (a^m)^n \].This rule states that you multiply the exponents together:\[ a^{m \times n} \].It streamlines expressions and helps in reducing complex exponents to simpler forms.Let's apply it to \[ (t^8)^2 \].We multiply the exponents, 8 and 2, to get:\[ t^{8 \times 2} = t^{16} \].This tells us that multiplying exponents is crucial when dealing with nested power situations.It's a simple yet powerful tool in expression simplification.
Simplification of Expressions
Simplification is our ultimate aim in handling mathematical expressions, making complex forms more manageable. Leveraging exponent rules is key in this process. After applying the power of a product rule and the power of a power rule, we combine the results for streamlined expressions.
  • First, ensure each component of your expression is fully simplified individually.
  • Calculate separately and then combine the results.
For the expression \[ (9t^8)^2 \],we simplified the components to\[ 81 \] and \[ t^{16} \],resulting in \[ 81t^{16} \].Simplifying transforms all parts into their simplest form for final, tidy results.

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Most popular questions from this chapter

The surface area of a cubical block of ice is represented by the polynomial \(6 x^{2}+36 x+54 .\) Use factoring to find an expression that represents the length of an edge of the block.

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