Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. $(x^{2} - x - 4)^{\frac{3}{4}} - 2 = 6$

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Isolate the term with the rational exponent by adding 2 to both sides: $ (x^2 - x - 4)^{\frac{3}{4}} = 8 $.

To eliminate the rational exponent, raise both sides of the equation to the reciprocal power, which is $ \frac{4}{3} $, so you get: $ \left((x^2 - x - 4)^{\frac{3}{4}}\right)^{\frac{4}{3}} = 8^{\frac{4}{3}} $.

Simplify the left side using the property of exponents $ (a^{m})^{n} = a^{mn} $, which results in $ x^2 - x - 4 = 8^{\frac{4}{3}} $.

Rewrite $ 8^{\frac{4}{3}} $ by expressing 8 as a power of 2 (since $8 = 2^3$), then apply the exponent: $ 8^{\frac{4}{3}} = (2^3)^{\frac{4}{3}} = 2^{3 \times \frac{4}{3}} = 2^4 $.

Now solve the quadratic equation $ x^2 - x - 4 = 2^4 $ by substituting $ 2^4 $ with its value and rearranging the equation to standard form: $ x^2 - x - (4 + 2^4) = 0 $. Then use the quadratic formula or factoring to find the values of $x$.

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Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, an exponent of 3/4 means raising the base to the third power and then taking the fourth root, or vice versa. Understanding how to manipulate and simplify expressions with rational exponents is essential for solving such equations.

Isolating the Expression

To solve equations involving rational exponents, first isolate the term with the exponent. This often involves adding or subtracting constants on both sides. Once isolated, you can raise both sides of the equation to the reciprocal power to eliminate the rational exponent.

Checking Solutions

When solving equations with rational exponents, extraneous solutions can arise, especially when both sides are raised to powers. Substituting proposed solutions back into the original equation ensures they satisfy the equation and are valid.

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x2−x−4)34−2=6(x^2 - x - 4)^{\(\frac{3}{4}\)} - 2 = 6