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Chapter 1: Problem 76

Perform each indicated operation. $$ \frac{5^{2}-7^{2}}{2(3+3)} $$

Short Answer

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Step by step solution

01

Simplify Powers

Calculate the powers in the numerator. First, find the value of each power: \(5^2 = 25\) \(7^2 = 49\)
02

Subtract Numerator Terms

Subtract the results found in Step 1 to simplify the numerator: \(25 - 49 = -24\)
03

Simplify the Denominator

Calculate the expression inside the parenthesis and then multiply by 2: \(3 + 3 = 6\) \(2(6) = 12\)
04

Divide the Numerator by the Denominator

Perform the division to find the final result: \(\frac{-24}{12} = -2\)

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Key Concepts

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

Exponents
To start solving the exercise, we first need to understand exponents. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression $$5^2$$, 5 is the base and 2 is the exponent. This means we multiply 5 by itself two times:
$$5 \times 5 = 25$$.
Similarly, for $$7^2$$, we calculate
$$7 \times 7 = 49$$.
Remember, always perform calculations involving exponents before proceeding to other operations in an expression.
Subtraction
After calculating the exponents, the next step is subtraction. Subtraction is a basic arithmetic operation where we take away one number from another.
In our example, after computing the exponents we have: $$25$$ and $$49$$.
We then subtract the smaller number from the larger number:
$$25 - 49 = -24$$.
Here, since we subtract a larger number from a smaller one, the result is negative, i.e., $$-24$$.
Parentheses
Expressions within parentheses must be solved first according to the order of operations. Parentheses group parts of expressions and dictate that calculations within them are completed before anything outside.
In our problem, we have the expression $$2(3+3)$$ in the denominator.
Calculate inside the parentheses first:
$$3 + 3 = 6$$.
Then proceed to multiply by the number outside the parentheses:
$$2 \times 6 = 12$$.
This gives us the denominator value.
Division
Finally, we reach division, the last operation to perform in our expression. Division distributes a numerator by a denominator.
In our case, the numerator is $$-24$$ and the denominator is $$12$$.
To perform the division, we simply divide $$-24$$ by $$12$$:
$$\frac{-24}{12} = -2$$.
The result of the division is $$-2$$.
Division is always one of the last operations done according to the order of operations (PEMDAS/BODMAS rules).

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Most popular questions from this chapter

Decide whether each statement is an example of the commutative, associative, identity, inverse, or distributive property. See Examples \(1,2,3,5,6,7,\) and \(9 .\) $$2(x+y)=2 x+2 y$$

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Simplify each expression. $$ -4-5(t-13) $$

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