Question

1. $3x^4 = 19683$
2. $x^3 - 10 = 1321$
3. $x^{7/11} cdot x^{9/5}$
4. $(y^{14})^{1/20}$

(3. $sqrt{49x^{94}y^{50}}$
(4. $sqrt{200x^{11}y^{40}z^{27}}$

Explanation:

Problem 9: Isolate $x^4$

Divide both sides by 3.
$\frac{3x^4}{3} = \frac{19483}{3}$
$x^4 = 6494.\overline{3}$

Problem 9: Solve for $x$

Take 4th root of both sides.
$x = \pm\sqrt[4]{6494.\overline{3}} = \pm9$
(Verification: $3\times9^4=3\times6561=19683$, which matches the original equation)

Problem 10: Isolate $x^3$

Add 10 to both sides.
$x^3 - 10 + 10 = 1321 + 10$
$x^3 = 1331$

Problem 10: Solve for $x$

Take cube root of both sides.
$x = \sqrt[3]{1331} = 11$

Problem 11: Apply exponent product rule

Add exponents when multiplying like bases.
$x^{\frac{7}{11} + \frac{9}{5}}$

Problem 11: Simplify the exponent

Find common denominator and add.
$\frac{7}{11} + \frac{9}{5} = \frac{35 + 99}{55} = \frac{134}{55}$
Result: $x^{\frac{134}{55}}$

Problem 12: Apply power of a power rule

Multiply the exponents.
$y^{4\times\frac{1}{20}}$

Problem 12: Simplify the exponent

Reduce the fraction.
$4\times\frac{1}{20} = \frac{1}{5}$
Result: $y^{\frac{1}{5}}$

Problem 13: Rewrite root as rational exponent

Square root = exponent $\frac{1}{2}$.
$(u^4x^{94}y^{50})^{\frac{1}{2}}$

Problem 13: Apply power of a product rule

Multiply each exponent by $\frac{1}{2}$.
$u^{4\times\frac{1}{2}}x^{94\times\frac{1}{2}}y^{50\times\frac{1}{2}}$

Problem 13: Simplify each exponent

Calculate each product.
$u^2x^{47}y^{25}$

Problem 14: Rewrite root as rational exponent

Square root = exponent $\frac{1}{2}$.
$(a^{100}x^{11}y^{10}z^{27})^{\frac{1}{2}}$

Problem 14: Apply power of a product rule

Multiply each exponent by $\frac{1}{2}$.
$a^{100\times\frac{1}{2}}x^{11\times\frac{1}{2}}y^{10\times\frac{1}{2}}z^{27\times\frac{1}{2}}$

Problem 14: Simplify each exponent

Calculate each product, rewrite fractional exponents.
$a^{50}x^{\frac{11}{2}}y^{5}z^{\frac{27}{2}} = a^{50}y^5x^5\sqrt{x}z^{13}\sqrt{z} = a^{50}x^5y^5z^{13}\sqrt{xz}$

Answer:

1. $x = \pm9$
2. $x = 11$
3. $x^{\frac{134}{55}}$
4. $y^{\frac{1}{5}}$
5. $u^2x^{47}y^{25}$
6. $a^{50}x^5y^5z^{13}\sqrt{xz}$ (or $a^{50}x^{\frac{11}{2}}y^5z^{\frac{27}{2}}$)