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Chapter 1: Problem 80

Evaluate using a calculator. $$ (t+4)^{2}-12 \div(19-17)+68, \text { for } t=5 $$

Short Answer

Expert verified
The evaluation of the expression is 143.

Step by step solution

01

- Substitute t with 5

Replace the variable t with the value 5 in the expression: $$ (t+4)^2 -12 梅 (19-17)+68 $$ becomes: Becomes $$ (5+4)^2 -12 梅 (19-17)+68
02

- Simplify the terms inside parentheses

First, simplify the expressions inside the parentheses: 5+4 = 9 and 19-17 = 2. This simplifies the expression to: $$ (9)^2 -12 梅 2 + 68 $$
03

- Evaluate the exponent

Calculate the square of 9: 9^2 = 81. Now the expression is: $$ 81 -12 梅 2 + 68 $$
04

- Perform division

Perform the division: 12 梅 2 = 6. The expression now becomes: $$ 81 - 6 + 68 $$
05

- Perform subtraction and addition

First, perform the subtraction: 81 - 6 = 75. Then, add 68 to 75: 75 + 68 = 143

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Key Concepts

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

Substitution
Substitution is one of the first steps in solving algebraic expressions. By substituting a variable with a specific value, you turn the expression into something you can directly evaluate.

For example, in our given exercise, we are asked to evaluate the expression for \( t = 5 \). Here, we replace every instance of \( t \) with 5:

\[ (t + 4)^2 - 12 梅 (19 - 17) + 68 \]

becomes

\[ (5 + 4)^2 - 12 梅 (19 - 17) + 68 \].

This simplification allows us to focus on arithmetic operations next.
Order of Operations
Understanding the order of operations is crucial in solving algebraic expressions. This sequence ensures you perform calculations in the right order. The acronym PEMDAS helps remember this order:

  • Parentheses
  • Exponents
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)
In our exercise, after substitution, we have:

\[ (5 + 4)^2 - 12 梅 (19 - 17) + 68 \]

Following PEMDAS, we first simplify inside the parentheses:

\[ (9)^2 - 12 梅 2 + 68 \].
Exponents
Exponents are numbers that indicate how many times to multiply the base by itself.

In our example, we next evaluate the exponent after simplifying inside the parentheses:

\[ (9)^2 - 12 梅 2 + 68 \].

The exponent here is 2, and it's applied to 9. So,

\[ 9^2 = 9 \times 9 = 81 \].

After applying the exponent, the expression becomes:

\[ 81 - 12 梅 2 + 68 \].
Parentheses
Parentheses in an expression indicate that the operations inside them should be performed first.

For our exercise, the parentheses help identify which parts of the expression to address initially:

\[ (5 + 4)^2 - 12 梅 (19 - 17) + 68 \].

Simplify inside the parentheses:

  • \( 5 + 4 = 9 \)
  • \( 19 - 17 = 2 \)
.

Now the expression becomes:

\[ (9)^2 - 12 梅 2 + 68 \].

Addressing parentheses first helps break down complex expressions into manageable steps.

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