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Chapter 1: Problem 89

Solve each equation. $$\left[(3+6)^{2} \div 3\right] \cdot 4=-54 x$$

Short Answer

Expert verified
The solution to the equation is \(x = -2\).

Step by step solution

01

Simplify the expression inside parenthesis

Firstly, the expression within the parentheses is simplified. This gives \((3 + 6) = 9\). So, the equation becomes \((9)^{2} \div 3\cdot4 = -54x\).
02

Solve Power

Next, take the square of 9 which gives \(81\). Substituting this back into the equation gives \(81\div3\cdot4= -54x\).
03

Perform Division

Now, perform the division 81 divided by 3 to get 27. The equation becomes \(27\cdot4 = -54x\).
04

Perform Multiplication

Next, multiply 27 by 4 to get 108. This simplifies to \(108 = -54x\).
05

Solve for x

Finally, divide both sides of the equation by -54 to solve for x. The equation is simplified to \(x = 108/-54\).

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Key Concepts

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

Parentheses Simplification
When solving equations, simplifying inside the parentheses is crucial. It helps in reducing the complexity of expressions by handling the terms enclosed first. For our exercise, we started by evaluating
  • \((3 + 6)\), which equals 9.
By simplifying the parentheses first, we bring clarity and can proceed accurately to the next steps. Remember, always prioritize parentheses to keep calculations neat and orderly. Parentheses can contain numbers, variables, or both, always simplify these first before applying other operations.
Exponents
Exponents, or powers, are shorthand for repeated multiplication. In our case,
  • \((9)^2\) means 9 multiplied by itself, which is 81.
Handling exponents effectively is key to simplifying expressions. Always resolve these after parentheses, according to the order of operations. Exponents change the magnitude of numbers quickly, so ensuring they are tackled correctly helps in achieving the right solution.
Division
Division helps in breaking down numbers into smaller parts. After simplifying exponents in the exercise, we moved to division,
  • \(81 \div 3 = 27\).
Division reduces an expression鈥檚 value smoothly and is crucial to balancing equations. It remains important to carry out division right after exponents when applicable, preserving the sequence of operations or PEMDAS/BODMAS.
Multiplication
Multiplication is combining numbers to form a larger product. In our exercise, after division, the next step was multiplying:
  • \(27 \cdot 4 = 108\).
Following multiplication with precision is key to maintaining the finesse of your calculations. Once simplified, multiplication provides a direct path toward solving equations and reaching correct answers effectively.

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Most popular questions from this chapter

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