Problem 89 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify the expression inside parenthesis

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

Step 2: Solve Power

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

Step 3: Perform Division

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

Step 4: Perform Multiplication

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

Step 5: Solve for x

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

Key Concepts

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’s 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.