Multiple choice

Give that $\displaystyle 10^{0.48}=x$ and $\displaystyle 10^{0.70}=y$ and $\displaystyle x^{z}=y^{2}$ then find the approximate value of z ?

  1. 2.9 approx

  2. 6.4 approx

  3. 3.0 approx

  4. 1.01 approx
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A Correct answer
Explanation

Given 10^0.48 = x and 10^0.70 = y. Then x^z = y^2 becomes (10^0.48)^z = (10^0.70)^2. This simplifies to 10^(0.48z) = 10^(1.40). Thus, 0.48z = 1.40, so z = 1.40 / 0.48 = 140 / 48 = 35 / 12 = 2.9166...

AI explanation

Substitute the values of x and y into the equation x raised to the power of z equals y squared to get 10 raised to the power of 0.48z equals 10 raised to the power of 1.40. Since the bases are the same, you can equate the exponents to form the equation 0.48z equals 1.40. Solving for z gives z equals 1.40 divided by 0.48, which simplifies to 140 divided by 48. Performing the division yields a result of approximately 2.9.

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