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Chapter 5: Problem 76

Simplify the given expression. \(9 .9-(-4.1)(8.5)^{2}\)

Short Answer

Expert verified
The simplified expression is 306.125.

Step by step solution

01

Simplify inside the parentheses

The expression given is \(9.9 - (-4.1)(8.5)^2\). Begin by acknowledging that within the parentheses, \(-4.1\) is a multiplier, and it affects the entire term \((8.5)^2\).
02

Calculate the exponentiation

Calculate \((8.5)^2\). To do this, simply multiply 8.5 by itself: \(8.5 \times 8.5 = 72.25\).
03

Handle the negative sign

Now, multiply \(-4.1\) by the result from Step 2. Therefore, \(-4.1 \times 72.25 = -296.225\).
04

Subtract from 9.9

Finally, perform the subtraction: \(9.9 - (-296.225)\). Subtracting a negative is the same as adding, so this becomes \(9.9 + 296.225 = 306.125\).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving numbers known as the base and the exponent. It is a way to express repeated multiplication. For example, in the expression \((8.5)^2\), the base is 8.5, and the exponent is 2.
**Calculating Exponents:**
To evaluate \((8.5)^2\), multiply 8.5 by itself:
  • The base 8.5 is multiplied by 8.5.
  • The result is 72.25.
Understanding exponents is crucial because it dictates the number of times the base is multiplied. In algebraic expressions, calculate the exponent before other operations without rearranging terms. This is part of the order of operations, which plays a vital role in simplifying expressions effectively.
Subtraction of Integers
Subtraction is one of the basic arithmetic operations. It involves taking away a value from another. When you subtract integers, it's like finding the difference between numbers.
**Handling Negative Numbers in Subtraction:**
When you see a negative sign, remember it's like reversing direction.
  • For example, in the expression \(9.9 - (-296.225)\), subtracting a negative number becomes adding its absolute value.
  • The expression becomes \(9.9 + 296.225\), simplifying down to 306.125.
This step involves changing subtraction of a negative to addition, which is essential for getting the right answer. Be mindful of signs to avoid mistakes!
Order of Operations
The order of operations is a rule that defines the sequence to follow when solving mathematical expressions. It tells you which parts of an expression to solve first to achieve the correct result.
**PEMDAS/BODMAS Rule:**
Here's an easy way to remember the order:
  • **P/B**: Parentheses/Brackets - Solve anything inside parentheses first.
  • **E/O**: Exponents/Orders - Next, compute exponents (or powers, roots).
  • **MD**: Multiplication and Division - Then perform any multiplication or division, left to right.
  • **AS**: Addition and Subtraction - Finally, handle any addition or subtraction, left to right.
Applying this rule is vital to simplifying expressions correctly, as it prevents errors by ensuring the operations are performed in the correct order. For the expression \(9.9 - (-4.1)(8.5)^2\), begin by handling the exponentiation part, followed by multiplication, and finally tackle the subtraction as per the order of operations.

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