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Chapter 2: Problem 47

Simplify using the order of operations. $$\frac{|-23+7|}{5^{2}-(-3)^{2}}$$

Short Answer

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Step by step solution

01

Simplify the expression inside the absolute value

First, simplify the expression inside the absolute value: \(|-23 + 7| = |-16|\). The absolute value of \(-16\) is \(16\).
02

Calculate the exponents

Next, simplify the exponents in the denominator: \(5^2 = 25\) and \((-3)^2 = 9\).
03

Subtract the squared terms

Subtract the result of \((-3)^2\) from \(5^2\): \(25 - 9 = 16\).
04

Simplify the fraction

Now, place the simplified numerator over the simplified denominator: \(\frac{16}{16} = 1\).

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Key Concepts

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

Absolute Value
The absolute value of a number is always non-negative. It's like measuring distance from zero on the number line. For example, if you have \(-16\), the absolute value is written as \(|-16|\) and simplifies to 16 since it's 16 units away from zero. Remember, absolute value removes any negative signs.
Exponents
Exponents represent repeated multiplication. For example, \(5^2\) means \(5 \times 5 = 25\). Note how the base (5) is multiplied by itself twice. Similarly, \((-3)^2\) means \((-3) \times (-3) = 9\). Here, even though -3 is negative, squaring it results in a positive product because a negative number times itself results in a positive number.
Fractions
A fraction consists of a numerator (top part) and a denominator (bottom part). For example, in \(\frac{16}{16}\), both the numerator and denominator are 16. To simplify fractions, you divide the numerator by the denominator. Here, dividing 16 by 16 simplifies to 1. Fractions follow the same math rules as whole numbers.
Simplification
Simplification makes expressions easier. Follow the order of operations, also known as PEMDAS \(Parentheses, Exponents, Multiplication and Division, Addition and Subtraction\). First, resolve the absolute value \(|-23 + 7|\) to get 16. Next, handle exponents: calculate \(5^2 = 25\) and \((-3)^2 = 9\). Subtract the exponent results: \(25 - 9 = 16\). Finally, simplify the fraction: \(\frac{16}{16} = 1\).

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Most popular questions from this chapter

Translate each English phrase to a mathematical expression. Then simplify. The difference of 70 and 98

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