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Apply the order of operations with real numbers

Interpreting numerical expressions with real numbers requires applying grouping symbols and exponents first, then multiplication and division from left to right, and finally addition and subtraction from left to right. This includes positive and negative integers, fractions, decimals, and familiar forms such as square roots, while distinguishing a negative sign from an exponent (for example, −3² versus (−3)²); the scope is numerical evaluation, not advanced symbolic manipulation or generalized algebraic proofs.

Detailed Explanation: Apply the order of operations with real numbers

To evaluate a numerical expression, follow this order:

  1. Evaluate inside grouping symbols such as parentheses or brackets.
  2. Evaluate exponents and square roots.
  3. Multiply and divide from left to right.
  4. Add and subtract from left to right.

Consider:

18−2[(−3)2−49]÷518-2\left[(-3)^2-4\sqrt{9}\right]\div 5

Step 1: Evaluate the exponent and square root inside the brackets.

(−3)2=9and9=3(-3)^2=9 \qquad \text{and} \qquad \sqrt{9}=3

So the expression becomes

18−2[9−4(3)]÷518-2[9-4(3)]\div 5

Notice that (−3)2=9(-3)^2=9 because the negative number is inside the parentheses. This is different from −32-3^2, which means −(32)=−9-(3^2)=-9.

Step 2: Multiply inside the brackets.

4(3)=124(3)=12

Now we have

18−2[9−12]÷518-2[9-12]\div 5

Step 3: Simplify inside the brackets.

9−12=−39-12=-3

The expression is now

18−2(−3)÷518-2(-3)\div 5

Step 4: Multiply and divide from left to right.

2(−3)=−62(-3)=-6 −6÷5=−65-6\div 5=-\frac{6}{5}

So the expression becomes

18−(−65)18-\left(-\frac{6}{5}\right)

Step 5: Subtract.

Subtracting a negative is adding:

18+65=905+65=965=19.218+\frac{6}{5}=\frac{90}{5}+\frac{6}{5}=\frac{96}{5}=19.2

Therefore,

19.2\boxed{19.2}

Learn by doing: Apply the order of operations with real numbers

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