Question

Solve for \(x\). $$ (x+3)^{4 / 3}=16 $$

Short answer

The solution to the equation is \(x = 5\)

Step-by-step solution

Eliminate Fractional Exponent

Raise both sides of the equation to the power of \(\frac{3}{4}\), in order to eliminate the fractional exponent from the left hand side. So, we will have \(((x+3)^{4 / 3})^{3 / 4} = 16^{3 / 4}\). Simplifying this results in \(x + 3 = 16^{3 / 4}\).

Simplify the Equation

Evaluate \(16^{3 / 4}\) to simplify the equation further. The fourth root of 16 is 2, and 2 cubed is 8. So, we will get \(x + 3 = 8\).

Solve for \(x\)

Subtract 3 from both sides to isolate \(x\). Therefore, \(x = 8 - 3 = 5\).

Key concepts

Fractional Exponents

Understanding fractional exponents is essential when solving exponential equations. A fractional exponent, such as \(4/3\), is another way of expressing a power and a root together. The numerator of the fraction signifies the exponent, indicating how many times the base is multiplied by itself, while the denominator signifies the root, which is essentially the inverse operation of raising a number to a power.

For example, when we see an expression like \(16^{3/4}\), it can be interpreted as taking the fourth root of 16 then raising the result to the third power. This is exactly what we did in our example exercise; we found that the fourth root of 16 is 2, and raising 2 to the third power gives us 8. It's important to break down these steps to simplify the process of dealing with fractional exponents.

Isolating Variables

The goal of isolating the variable is to get the variable on one side of the equality, 'alone'. This is a pivotal step in solving any algebraic equation, including exponential ones. To do this, we often perform inverse operations on both sides of the equation to maintain balance.

In the given example, once we simplified \(16^{3/4}\) to 8, we were left with the expression \(x + 3 = 8\). To isolate \(x\), we subtracted 3 from both sides, resulting in \(x = 8 - 3\), which simplifies to \(x = 5\). This step is significant because it gives us the value of \(x\) that satisfies the original equation. Each operation we perform gets us closer to the solution, but we must apply these operations evenly to both sides to uphold the equation's integrity.

Exponent Properties

Exponent properties, or laws of exponents, are rules that guide us on how to handle expressions with exponents efficiently. These rules become incredibly helpful when dealing with equations involving exponential terms. One such rule that applies to our example is the power of a power property, which states that \( (a^m)^n = a^{m \times n} \).

When we raised both sides of \( (x+3)^{4/3} \) by \(3/4\) in our exercise, we were effectively using this property. By raising the power to another power, we multiply the exponents, resulting in \( (x+3)^{4/3 \times 3/4} \) which simplifies to \( (x+3)^{1} \) or just \(x+3\). Understanding this property allowed us to remove the fractional exponent and simplify the equation to achieve a solvable form for \(x\).