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Chapter 4: Problem 44

Evaluate each expression if \(a=2, b=4,\) and \(c=-3\) $$3(b-1)^{4}$$

Short Answer

Expert verified
The expression evaluates to 243.

Step by step solution

01

Substitute the Given Values

Start by substituting the given values into the expression. Replace \(b\) with \(4\) since \(b = 4\). The expression becomes: \[ 3(4-1)^{4} \]
02

Simplify Inside the Parentheses

Simplify the expression inside the parentheses by performing the subtraction: \(4 - 1 = 3\). Now, the expression is: \[ 3(3)^{4} \]
03

Evaluate the Exponent

Calculate the value of \(3^{4}\). This means multiplying 3 by itself four times: \(3 imes 3 imes 3 imes 3 = 81\). Now the expression is: \[ 3 imes 81 \]
04

Perform the Final Multiplication

Multiply the result by 3 to complete the evaluation: \(3 imes 81 = 243\). So, the final value of the expression is 243.

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Key Concepts

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

Substitution
Substitution is an essential first step in evaluating algebraic expressions. It involves replacing variables with their given values. This is like decoding a message by translating each letter into its corresponding symbol.
When you're given values for variables, like in the exercise
  • Replace each variable with the number it represents. For instance, if you have the expression with 'b' and know that \(b=4\), substitute 'b' with '4'.
In our specific expression, we substitute 'b' with 4 into \(3(b-1)^4\) to get \(3(4-1)^4\).
Remember, one small mistake in substitution can lead to an incorrect solution, so it's crucial to be accurate and meticulous in this step.
Order of Operations
The order of operations is a fundamental principle in mathematics used to clarify which procedures to perform first in a given expression. If operations are not performed in a systematic order, the final result can be incorrect. Here's the basic order you should follow, often remembered by the acronym PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)
For the given expression \(3(b-1)^4\), we first simplify inside the parentheses, then evaluate the exponent and finally, multiply by 3.
Sticking to this order ensures consistency and correctness across various expressions. This systematic approach is a lifesaver in complex calculations!
Exponents
Exponents are a shorthand way to represent repeated multiplication of the same number. When you see \(b^n\), it means you multiply 'b' by itself 'n' times. Exponents make calculations more concise and manageable.
In the exercise, after substitution and simplifying inside the parentheses, you have \(3^4\). This means:
  • Multiply 3 by itself a total of 4 times: \(3 \times 3 \times 3 \times 3 = 81\)
Understanding exponents is vital, as they appear frequently in mathematics. Always handle them with care, as small changes in the base or exponent can significantly impact the result. The exponential step is evaluate before performing any other operations outside.

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