Solving equations Solve the following equations.


3(ˣ³⁻⁴) = 15

Verified step by step guidance

1. Step 1: Start by isolating the exponential expression. Divide both sides of the equation by 3 to simplify: \( \frac{3(x^3 - 4)}{3} = \frac{15}{3} \).
2. Step 2: Simplify the equation from Step 1. This results in \( x^3 - 4 = 5 \).
3. Step 3: Solve for \( x^3 \) by adding 4 to both sides of the equation: \( x^3 = 5 + 4 \).
4. Step 4: Simplify the right side of the equation from Step 3: \( x^3 = 9 \).
5. Step 5: Solve for \( x \) by taking the cube root of both sides: \( x = \sqrt[3]{9} \).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as x in the expression a^x. To solve these equations, one often uses logarithms to isolate the variable. Understanding the properties of exponents and logarithms is crucial for manipulating and solving these types of equations.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable on one side and all other terms on the opposite side. In the given equation, this means simplifying and dividing to find the value of x.

Properties of Equality

The properties of equality state that if two expressions are equal, then one can be manipulated without changing the equality. This includes adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number. These properties are essential for solving equations systematically.