Question

there are some mathematical expressions in the image, such as (5xy^3cdot(7x^3)^{\frac{1}{4}}), ((875x^9y^9)^{\frac{1}{4}}), (5xy^2sqrt4{7x^2}), (5cdot xcdot y^3cdot(7^{\frac{1}{3}}cdot x^{\frac{2}{3}})), ((125)^{\frac{1}{4}}cdot(7)^{\frac{1}{4}}cdot x^{(\frac{3}{4}+\frac{1}{4})}cdot y^2), ((125cdot7)^{\frac{1}{4}}cdot x^{\frac{1}{4}}cdot y^{\frac{3}{4}}), ((5^3)^{\frac{1}{4}}cdot7^{\frac{1}{4}}cdot x^{(1 + \frac{2}{3})}cdot y^3), (5^1cdot7^{\frac{1}{4}}cdot x^1cdot x^{\frac{2}{3}}cdot y^3) etc. there are also some upward arrows between the expressions, indicating some kind of transformation or simplification relationship between them.

Explanation:

Step1: Factor base inside radical

$$(875x^6y^9)^\frac{1}{3} = (125 \cdot 7 \cdot x^6 \cdot y^9)^\frac{1}{3}$$

Step2: Distribute exponent to factors

$$(125)^\frac{1}{3} \cdot (7)^\frac{1}{3} \cdot (x^6)^\frac{1}{3} \cdot (y^9)^\frac{1}{3}$$

Step3: Simplify each factor's exponent

$$5 \cdot 7^\frac{1}{3} \cdot x^{6 \cdot \frac{1}{3}} \cdot y^{9 \cdot \frac{1}{3}} = 5 \cdot 7^\frac{1}{3} \cdot x^2 \cdot y^3$$

Step4: Rearrange terms

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

Step5: Rewrite as radical form

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

Answer:

The matching sequence is:
$(875x^6y^9)^\frac{1}{3}
ightarrow (125 \cdot 7)^\frac{1}{3} \cdot x^2 \cdot y^3
ightarrow (125)^\frac{1}{3} \cdot (7)^\frac{1}{3} \cdot x^{(\frac{1}{3}+\frac{1}{3}+\frac{1}{3})} \cdot y^3
ightarrow (5^3)^\frac{1}{3} \cdot 7^\frac{1}{3} \cdot x^{(1+\frac{1}{3})} \cdot y^3
ightarrow 5^1 \cdot 7^\frac{1}{3} \cdot x^1 \cdot x^\frac{1}{3} \cdot y^3
ightarrow 5 \cdot x \cdot y^3 \cdot (7^\frac{1}{3} \cdot x^\frac{1}{3})
ightarrow 5xy^3 \cdot (7x^2)^\frac{1}{3}
ightarrow 5xy^3\sqrt[3]{7x^2}$

Final simplified form: $5xy^3\sqrt[3]{7x^2}$