Complex Expressions

Grade 9 · algebra · 100 practice problems · read aloud

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Complex Expressions: Putting It All Together

Complex expressions combine multiple operations like addition, subtraction, multiplication, division, and exponents, often with parentheses. Mastering them is essential for solving real-world problems and preparing for advanced algebra. 🧩

Step-by-Step Guide: The Order of Operations

Always follow PEMDAS to get the correct answer:

  1. Parentheses: Simplify expressions inside grouping symbols first.
  2. Exponents: Evaluate all powers (e.g., 3²).
  3. Multiplication & Division: Work from left to right.
  4. Addition & Subtraction: Work from left to right.

Worked Examples

Example 1: Simplify \( 10 + (8 - 2^2) \div 2 \)

  1. Parentheses & Exponents: Inside the parentheses, handle the exponent first: \( 8 - 2^2 = 8 - 4 \). Then subtract: \( 8 - 4 = 4 \).
    Expression is now: \( 10 + 4 \div 2 \)
  2. Division: \( 4 \div 2 = 2 \).
    Expression is now: \( 10 + 2 \)
  3. Addition: \( 10 + 2 = 12 \).

Answer: 12

Example 2: Simplify \( 5 \times [6 + (3^2 - 8)] \)

  1. Inner Parentheses & Exponent: \( 3^2 - 8 = 9 - 8 = 1 \).
    Expression is now: \( 5 \times [6 + 1] \)
  2. Brackets (Grouping): \( 6 + 1 = 7 \).
    Expression is now: \( 5 \times 7 \)
  3. Multiplication: \( 5 \times 7 = 35 \).

Answer: 35

Common Mistakes to Avoid ⚠️

  • Ignoring PEMDAS Order: Don't just go left to right! Multiplication does NOT always come before division; work them left to right. The same is true for addition and subtraction.
  • Misreading Negative Signs: Remember that \( -3^2 = -9 \) because the exponent applies only to the 3 (\( -(3 \times 3) \)).
  • Rushing Through Steps: Rewrite the expression after each simplification step to avoid losing track.

Tips & Tricks

  • Use the mnemonic "Please Excuse My Dear Aunt Sally" to remember PEMDAS.
  • Treat fractions as implied parentheses: simplify the numerator and denominator separately before dividing.
  • When in doubt, add more parentheses to clarify the order you intend.

How to Practice

To build confidence, try these activities:

  • Create your own expressions and solve them.
  • Use online math practice websites for instant feedback.
  • Work with a partner and explain your steps out loud—teaching someone else is the best way to learn!
  • Start with simple 3-step problems and gradually increase the complexity.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 √(16) + 3² - 2³ = ?

Hint: Evaluate each term separately, starting with roots and exponents, then combine the results using the correct order of operations.

Show the answer

Answer: 5

  1. Evaluate the square root** √(16) means the principal (positive) square root of 16. √(16) = 4 **
  2. Evaluate the exponent 3²** 3² = 3 × 3 = 9 **
  3. Evaluate the exponent 2³** 2³ = 2 × 2 × 2 = 8 **
  4. Substitute back into the expression** 4 + 9 - 8 **
  5. Perform addition and subtraction from left to right** 4 + 9 = 13 13 - 8 = 5 **Final Answer:** 5

Let's solve step by step. We have: √(16) + 3² - 2³ **

2 2³ × (4 + 3) - √49 = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication/division from left to right, and finally addition/subtraction from left to right.

Show the answer

Answer: 49

  1. Handle the exponent** 2³ means 2 × 2 × 2 = 8 So now we have: 8 × (4 + 3) - √49 **
  2. Simplify inside parentheses** 4 + 3 = 7 So now: 8 × 7 - √49 **
  3. Handle the square root** √49 = 7 (because 7 × 7 = 49) So now: 8 × 7 - 7 **
  4. Perform multiplication** 8 × 7 = 56 So now: 56 - 7 **
  5. Perform subtraction** 56 - 7 = 49 **Final Answer:** 49

Let's solve step-by-step. We have: 2³ × (4 + 3) - √49 **

3 2³ × (4 - 1)² ÷ √36 = ?

Hint: Follow the order of operations: evaluate exponents first, then parentheses, then multiplication and division from left to right.

Show the answer

Answer: 12

  1. Handle the exponent 2³** 2³ means 2 × 2 × 2 = 8 So now we have: 8 × (4 - 1)² ÷ √36 **
  2. Simplify inside parentheses** (4 - 1) = 3 So now: 8 × (3)² ÷ √36 **
  3. Square the 3** (3)² = 9 So now: 8 × 9 ÷ √36 **
  4. Square root of 36** √36 = 6 So now: 8 × 9 ÷ 6 **
  5. Multiply and divide left to right** First: 8 × 9 = 72 Then: 72 ÷ 6 = 12 **Final Answer:** 12

Let's solve step by step. We have: 2³ × (4 - 1)² ÷ √36 **

4 log₂(8) + 3² - √(25) = ?

Hint: Evaluate each term separately using the definitions of logarithm, exponent, and square root. Remember that log base 2 of a number asks '2 to what power equals this number?'

Show the answer

Answer: 7

  1. Evaluate log₂(8). Since 2³ = 8, log₂(8) = 3
  2. Evaluate 3². 3 × 3 = 9
  3. Evaluate √(25). The square root of 25 is 5
  4. Substitute the values: 3 + 9 - 5
  5. Perform addition: 3 + 9 = 12
  6. Perform subtraction: 12 - 5 = 7

The answer is 7.

5 log₂(32) + 3² - √(81) = ?

Hint: Break down each term: evaluate the logarithm, then the exponent, then the square root, and finally combine them using the correct order of operations.

Show the answer

Answer: 5

  1. Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5.
  2. Evaluate 3². 3 × 3 = 9.
  3. Evaluate √(81). Since 9 × 9 = 81, √(81) = 9.
  4. Substitute the values back into the expression: 5 + 9 - 9.
  5. Perform the operations from left to right: 5 + 9 = 14, then 14 - 9 = 5. The final answer is 5.

6 log₃(27) + 2⁴ - √(169) = ?

Hint: Break the expression into parts: evaluate the logarithm, the exponent, and the square root separately before combining them.

Show the answer

Answer: 6

  1. Evaluate log₃(27). Since 3^3 = 27, log₃(27) = 3.
  2. Evaluate 2⁴. 2 × 2 × 2 × 2 = 16.
  3. Evaluate √(169). Since 13 × 13 = 169, √(169) = 13.
  4. Substitute the values back into the expression: 3 + 16 - 13.
  5. Perform the addition: 3 + 16 = 19.
  6. Perform the subtraction: 19 - 13 = 6.

The answer is 6.

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