Problem 90 - Algebra and Trigonometry

Question

Solve each equation. $$2^{3}-\left[4(5-3)^{3}\right]=-8 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\).

Step 1: Simplification

Initially, simplify the expression on the left side of the equation, So, \(2^{3}-\left[4(5-3)^{3}\right] = 8 - \left[4*2^{3}\right] = 8 - 4*8 = 8 - 32 = -24.\n\nThe equation now becomes: -24 = -8x

Step 2: Division

Isolate \(x\) by dividing all terms by -8,So, \((-24/-8) = x\)

Step 3: Simplification

Simplify the right side of the equation,So, \(x = 3\)

Key Concepts

Exponents

Understanding exponents is crucial when solving exponential equations. An exponent, also known as a power, represents how many times a number, called the base, is multiplied by itself. For instance, in the expression \(2^3\), the number 2 is the base and 3 is the exponent, meaning that 2 is multiplied by itself 3 times: \(2 \times 2 \times 2 = 8\).

When solving equations with exponents, remember common rules like \(a^{m} \times a^{n} = a^{m+n}\) and \(\left(a^{m}\right)^{n} = a^{m \times n}\). Always calculate the value of the exponents first before proceeding with any other operations. This adherence to the correct order of operations ensures that the simplification process is accurate and the solution found is correct.

Order of Operations

The order of operations dictates the sequence in which parts of a mathematical expression should be simplified. To correctly solve an equation, you should follow the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the given problem, operations within brackets and exponents are tackled first, followed by multiplication, and finally, subtraction. This step-by-step approach helps you avoid errors and ensures the correct simplification of the equation. Always pay close attention to brackets and perform their enclosed operations ahead of others, just as how the textbook solution handled \(4(5-3)^3\) by first addressing the subtraction within the parentheses.

Isolating Variables

To find the value of an unknown variable, it's necessary to isolate it on one side of the equation. This is commonly done using inverse operations. Inverse operations are operations that 'undo' each other, like addition and subtraction or multiplication and division.

In our exercise, we isolate \(x\) by dividing both sides of the equation by -8. This technique effectively reverses the multiplication of \(x\) with -8, leaving \(x\) alone on one side of the equation. It's essential to perform the same operation on both sides to maintain the balance of the equation. This balance means that if an equation is true before an operation is applied, it will remain true afterward.

Equation Simplification

Equation simplification refers to the process of breaking down complex expressions into simpler forms to make it easier to solve. Simplifying an equation involves combining like terms, reducing fractions, or simplifying powers, as necessary. It's a crucial step before isolating the variable, as it makes the rest of the problem easier to manage.

In our example, you observed that after simplifying the exponents and the operations within brackets, the left side of the equation was reduced from \(2^3 - [4(5-3)^3]\) to -24. This simplification paved the way to isolate \(x\), showcasing the importance of simplifying an equation thoroughly before proceeding to solve for the unknown variable.