Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x - 4)^2/3 = 16

Verified step by step guidance

1. Recognize that the equation involves a rational exponent: \( (x - 4)^{\frac{2}{3}} = 16 \). The exponent \(\frac{2}{3}\) means we first take the cube root of \(x - 4\) and then square the result.
2. To eliminate the rational exponent, raise both sides of the equation to the reciprocal power, which is \(\frac{3}{2}\). This gives: \(\left((x - 4)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 16^{\frac{3}{2}}\).
3. Simplify the left side using the property of exponents \(\left(a^{m}\right)^{n} = a^{mn}\), so the left side becomes \(x - 4\). The equation now is \(x - 4 = 16^{\frac{3}{2}}\).
4. Evaluate \(16^{\frac{3}{2}}\) by first finding the square root of 16, which is 4, and then raising 4 to the 3rd power, which is \$4^3$. This gives the right side value.
5. Solve for \(x\) by adding 4 to both sides: \(x = 16^{\frac{3}{2}} + 4\). Finally, check all proposed solutions by substituting them back into the original equation to ensure they satisfy it.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers combined; for example, an exponent of m/n means taking the nth root and then raising to the mth power. Understanding how to manipulate expressions with rational exponents is essential for solving equations like (x - 4)^(2/3) = 16.

Isolating the Variable

To solve equations, isolate the term containing the variable by undoing operations step-by-step. For rational exponents, this often involves raising both sides of the equation to the reciprocal power to eliminate the exponent and solve for the variable.

Checking Solutions

After solving, substitute proposed solutions back into the original equation to verify they satisfy it. This is crucial when dealing with rational exponents, as raising to even roots can introduce extraneous solutions that do not satisfy the original equation.