Question

drag each tile to the correct box.
order the simplification steps of the expression below using the properties of rational exponents.
$sqrt3{875x^{5}y^{9}}$
$5xy^{3}cdot(7x^{2})^{\frac{1}{3}}$
$5^{\frac{1}{3}}cdot7^{\frac{1}{3}}cdot x^{1}cdot x^{\frac{2}{3}}cdot y^{3}$
$(125cdot7)^{\frac{1}{3}}cdot x^{\frac{5}{3}}cdot y^{3}$
$(875x^{5}y^{9})^{\frac{1}{3}}$
$5cdot xcdot y^{3}cdot(7^{\frac{1}{3}}cdot x^{\frac{2}{3}})$
$5xy^{3}sqrt3{7x^{2}}$
$(125)^{\frac{1}{3}}cdot(7)^{\frac{1}{3}}cdot x^{(1+\frac{2}{3})}cdot y^{3}$
$(5^{3})^{\frac{1}{3}}cdot7^{\frac{1}{3}}cdot x^{(1+\frac{2}{3})}cdot y^{3}$

Explanation:

Step1: Start with original radical

$\sqrt[3]{875x^5y^9}$

Step2: Convert radical to rational exponent

$(875x^5y^9)^\frac{1}{3}$

Step3: Factor 875, split exponents

$(125 \cdot 7)^\frac{1}{3} \cdot x^\frac{5}{3} \cdot y^\frac{9}{3}$

Step4: Rewrite $x^\frac{5}{3}$, simplify $y^\frac{9}{3}$

$(125)^\frac{1}{3} \cdot (7)^\frac{1}{3} \cdot x^{(1+\frac{2}{3})} \cdot y^3$

Step5: Rewrite 125 as $5^3$

$(5^3)^\frac{1}{3} \cdot 7^\frac{1}{3} \cdot x^{(1+\frac{2}{3})} \cdot y^3$

Step6: Simplify exponents, split $x$ term

$5^1 \cdot 7^\frac{1}{3} \cdot x^1 \cdot x^\frac{2}{3} \cdot y^3$

Step7: Group constant/variable terms

$5 \cdot x \cdot y^3 \cdot (7^\frac{1}{3} \cdot x^\frac{2}{3})$

Step8: Combine grouped terms into single exponent

$5xy^3 \cdot (7x^2)^\frac{1}{3}$

Step9: Convert back to radical form

$5xy^3\sqrt[3]{7x^2}$

Answer:

1. $\sqrt[3]{875x^5y^9}$
2. $(875x^5y^9)^\frac{1}{3}$
3. $(125 \cdot 7)^\frac{1}{3} \cdot x^\frac{5}{3} \cdot y^\frac{9}{3}$
4. $(125)^\frac{1}{3} \cdot (7)^\frac{1}{3} \cdot x^{(1+\frac{2}{3})} \cdot y^3$
5. $(5^3)^\frac{1}{3} \cdot 7^\frac{1}{3} \cdot x^{(1+\frac{2}{3})} \cdot y^3$
6. $5^1 \cdot 7^\frac{1}{3} \cdot x^1 \cdot x^\frac{2}{3} \cdot y^3$
7. $5 \cdot x \cdot y^3 \cdot (7^\frac{1}{3} \cdot x^\frac{2}{3})$
8. $5xy^3 \cdot (7x^2)^\frac{1}{3}$
9. $5xy^3\sqrt[3]{7x^2}$