Questions: DeltaMath Student Application deltamath.com/app/student/3499378/26085724/9747ab50f190040e6725690a2a7b4d11 DeltaMath Back to Home Exponent Properties and Rational Exponents Due: January 12 at 11:59 PM Grade: 77% (19^5)^2=19^10 (19^(1/5))^5=19 Let x=19^(1/5). Rewrite the equation in the previous answer box, replacing 19^(1/5) with x (substitute). Exponents: Expand/Condense (Monomial Base) Intro to Fractional Exponents (Guided) Exponential and Radical Form

Solution Steps

The problem provides two equations involving exponents:

1. \(\left(19^{5}\right)^{2} = 19^{10}\)
2. \(\left(19^{\frac{1}{5}}\right)^{5} = 19\)

The first equation demonstrates the power of a power property, where \(\left(a^{m}\right)^{n} = a^{m \cdot n}\). The second equation shows that raising a number to a fractional exponent and then to the reciprocal of that exponent returns the original number.

The problem asks to rewrite the second equation by substituting \( x = 19^{\frac{1}{5}} \). The second equation is: \[ \left(19^{\frac{1}{5}}\right)^{5} = 19 \] Substituting \( x = 19^{\frac{1}{5}} \), the equation becomes: \[ x^{5} = 19 \]

The substitution \( x = 19^{\frac{1}{5}} \) implies that \( x^{5} = \left(19^{\frac{1}{5}}\right)^{5} = 19 \). This confirms that the substitution is correct and the rewritten equation is valid.

Final Answer