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MathematicsAlgebra

Given that $5^{0.6} = x, 5^{0.2} = y$ and $x^{z} = y^{2}$ , then the value of z is:

2/3

3/4

4/5

5/6

Correct Answer

📐 MA • Math Exponent Power RuleMathematicsAlgebra

Option A

2/3

Quick Summary:

Express both $x$ and $y$ as powers of $5$ , then equate the exponents directly by ignoring the base 5.

📐MAMath SolutionExponent Power Rule

📋 Given

Given $x = 5^{0.6}$ and $y = 5^{0.2}$ , with the equation $x^{z} = y^{2}$ .

🔢 Formula Used

(am)n=am×n(a^m)^n = a^{m \times n}(am)n=am×n

⚡ Exam Hall Shortcut / Speed Trick

Express both $x$ and $y$ as powers of $5$ , then equate the exponents directly by ignoring the base 5.

⚠️ Common Student Trap / Pitfall

Confusing $0.6$ and $0.2$ with fractions or failing to multiply the power of $y$ by $2$ , which leads to $z=0.4$ instead of $z=0.666...$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Convert x and y to powers of 5

Substitute the given values of $x$ and $y$ into the equation $x^{z} = y^{2}$ .

(50.6)z=(50.2)2(5^{0.6})^{z} = (5^{0.2})^{2}(50.6)z=(50.2)2

Apply power of power rule

Using $(a^m)^n = a^{m \times n}$ , expand both sides.

50.6z=50.45^{0.6z} = 5^{0.4}50.6z=50.4

Equate the exponents

Since the bases are equal, the exponents must be equal: $0.6z = 0.4$ .

z=0.40.6=46=23z = \frac{0.4}{0.6} = \frac{4}{6} = \frac{2}{3}z=0.60.4​=64​=32​

✅

A is correct because solving the equation $5^{0.6z} = 5^{0.4}$ yields $z = 2/3$ .

Core Concepts Used

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💡 EXAM TIP

This concept is fundamental to solving logarithmic equations and base-change problems in quantitative aptitude exams.

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