The expression 4sqrt[4]{5^3y^(20)} * sqrt[4]{3^3y^2} is equivalent to cy^d, where c and d are positive constants and y gt 1....

1. TRANSLATE the radical notation to workable form

• Given: \(4\sqrt[4]{5^3y^{20}} \cdot \sqrt[4]{3^3y^2}\)
• TRANSLATE to fractional exponents: \(4 \cdot (5^3y^{20})^{1/4} \cdot (3^3y^2)^{1/4}\)

This conversion is essential because fractional exponents follow standard algebraic rules, making the expression much easier to manipulate.

2. SIMPLIFY using exponent rules

• Apply \((x^m)^n = x^{mn}\) to each radical term:
  • \((5^3y^{20})^{1/4} = 5^{3/4} \cdot y^{20/4} = 5^{3/4} \cdot y^5\)
  • \((3^3y^2)^{1/4} = 3^{3/4} \cdot y^{2/4} = 3^{3/4} \cdot y^{1/2}\)
• Expression becomes: \(4 \cdot 5^{3/4} \cdot y^5 \cdot 3^{3/4} \cdot y^{1/2}\)

3. SIMPLIFY by combining like bases

• Use \((ab)^n = a^n \cdot b^n\) in reverse: \(5^{3/4} \cdot 3^{3/4} = (5\cdot3)^{3/4} = 15^{3/4}\)
• Use \(x^m \cdot x^n = x^{m+n}\): \(y^5 \cdot y^{1/2} = y^{5.5}\)
• Result: \(4 \cdot 15^{3/4} \cdot y^{5.5}\)

4. SIMPLIFY the coefficient calculation

• Calculate \(15^{3/4} = (15^3)^{1/4} = (3375)^{1/4} \approx 7.62\) (use calculator)
• So \(c = 4 \times 7.62 = 30.5\) and \(d = 5.5\)
• Therefore: \(c + d = 30.5 + 5.5 = 36.0\)

Answer: C) 36.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert radical notation to fractional exponents, particularly with the fourth root symbol ∜.

They might try to work directly with the radicals or incorrectly convert \(\sqrt[4]{x^3}\) to \(x^3\) instead of \(x^{3/4}\), making the problem much more complicated than necessary. Without this crucial first step, they get bogged down in messy radical arithmetic and abandon systematic solution.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly convert to fractional exponents but make arithmetic errors when combining exponents, especially with fractions.

Common mistakes include: \(y^{20/4} \cdot y^{2/4} = y^{22/4}\) instead of \(y^{22/4}\), or miscalculating \(5 + 1/2 = 5.2\) instead of \(5.5\). These errors cascade through to the final calculation of \(c + d\).

This may lead them to select Choice A (30.5) or Choice B (32.0).

The Bottom Line:

This problem tests your ability to systematically convert and manipulate expressions with radicals and fractional exponents. Success depends on methodical application of exponent rules rather than trying shortcuts.